This, only their second win in 18 Premiership matches, takes Hammers one place and two points above Albion, who drop to second from bottom of the Premiership, just ahead of basement side Sunderland.
Sinclair, recalled to the side in place of the suspended Joe Cole, showed the appetite for the battle that has been lacking for much of this season.
He was guilty of missing a great chance to give the visitors the lead but made up for that glaring first half miss by scoring with his left foot from a tight angle on the stroke of half-time.
The Baggies equalised five minutes into the second period through Danny Dichio but a rare West Ham counter-attack ended gloriously for Sinclair, who headed home from eight yards after Les Ferdinand's header rebounded back to him off the crossbar in the 66th minute.
West Brom must wonder how they didn't take at least a point from this absorbing cliff-hanger. They had two goals controversially disallowed for offside, a reasonable penalty appeal turned down when Rufus Brevett's foul was deemed to have been committed on the 18-yard line and also hit the woodwork on a day of misery for the Black Country club.
With Dichio and Lee Hughes a constant handful for Tomas Repka and Ian Pearce, their direct, rugged, route one approach created plenty of pressure, especially in the second half.
Hughes (twice) and substitute Dobie both found themselves in gilt-edged one-on-one situations but on all three occasions the brilliant James came off his line to deny the home side. How valuable those saves could prove as the relegation battle continues to unfold over the remaining 10 games.
Albion no doubt considered themselves very unfortunate to be behind at half-time, having hit the post after just seven minutes and then seen a perfectly good goal disallowed for offside midway through the first period.
When Jason Koumas floated in a free-kick from the left, Darren Moore climbed higher than Pearce to send a thumping header against James' left upright.
After that lucky escape Hammers were relieved to see the linesman's flag raised when Nigerian international Udeze played a one-two on the left and prodded the return pass under the advancing James and into the net. Although Hughes was arguably in an offside position in the middle, he played no active part in the move and the strike from Udeze should have counted.
West Ham improved as a nervous, scrappy first half progressed. On the half-hour mark they produced their best move when Paolo Di Canio played in the overlapping Steve Lomas, whose byeline cross found Lee Bowyer, although his effort was deflected wide of the far post for a corner.
Despite riding their luck early on, Hammers missed a golden chance to take the lead in the 38th minute when Sinclair hit his follow up shot into the ground and up onto the bar from just six yards after Russell Hoult had spilled Michael Carrick's 20-yard strike.
But on the stroke of half-time Sinclair made amends for that bad miss. Di Canio laid the ball back for Lomas whose first-time, left-footed cross was met at the near post by Sinclair. He took one touch with his right foot and hit a shot with his left that nestled inside the far post.
On a day when only 100 per cent effort was acceptable, Glenn Roeder bravely decided to withdraw his skipper, Di Canio, just three minutes into the second half, much to the Italian's obvious disgust. Di Canio hadn't been very competitive in the first half, even uncharacteristically conceding possession too often, so Jermain Defoe joined Ferdinand up front.
Within two minutes, though, another inswinging Koumas cross from the left was met in the air by Dichio, who climbed on the shoulders of Repka to nod Albion level from the edge of the six-yard box.
Dichio thought he had given West Brom the lead just six minutes later. Hughes' pass to him looked squarish but, once again, the linesman came to Hammers' rescue to deny the goal with another offside flag. The tension and anxiety in the visitors' defence then spilled over into a verbal slanging match between James and Repka but after a bit of pushing and shoving peace was restored.
Just as well that the keeper did regain his composure, because he would be tested to the limit as Albion pumped endless balls high into the penalty area. Hammers had their backs to the wall for long periods but made the most of the only goalscoring chance that came their way.
When Hoult miscued a clearance, Hammers played the ball wide to the right to Lee Bowyer, whose excellent deep cross was met at the far post by Defoe. His sidefooted volley to Ferdinand six yards out looked to be on its way in but the big striker - who worked hard at both ends - thumped his header against the bar...only for Sinclair to nod it back into the net with Hoult still grounded.
Could Hammers hang on? The rest of this titanic battle was the David James Show. His three heroic saves in the last 20 minutes not only rescued his fellow defenders, they have now given West Ham's survival hopes a massive boost on a day when they simply couldn't afford to lose.
By Tony McDonald
Minute-by-minute:
5 - Hammers create the first real chance of the game, as Sinclair's superb crossfield pass from left to right finds Di Canio in space and the Italian attempts to find the run of Bowyer with a chipped cross that is cut out by Hoult.
7 - The hosts reply immediately and go even closer to taking the lead as Koumas' deep free-kick from the left is met by Moore, who crashes a header against the post. Johnson gratefully clears the loose ball as it bounces in the six-yard box.
18 - This time Hammers go close from a free-kick on the left, as Di Canio's effort is met at the far post by Ferdinand, who heads high over the bar from eight yards out.
21 - After Pearce is penalised for a foul on Dichio some 25 yards out, Koumas curls the free-kick just wide of the target.
24 - A scare for Hammers as Udeze runs on to Dichio's pass and slips the ball past James into the net, but the goal is thankfully ruled out for offside.
26 - Another Di Canio free-kick is this time chipped into the path of Bowyer, whose header from 12 yards out flies just wide.
29 - Lomas skips on to Di Canio's pass on the left and races into the area before cutting the ball back for Bowyer, whose goalbound effort is agonisingly deflected wide for a corner.
36 - Gregan booked for a foul on Ferdinand.
38 - Hammers waste an absolutely glorious chance to open the scoring, as Carrick's fierce low shot is parried back out to Sinclair who, six yards out, has the goal at his mercy but amazingly fires his snap-shot into the ground and watches in horror as the ball bounces up and clips the crossbar before dropping over for a goal kick.
43 - Wallwork's low shot from the edge of the penalty area is comfortably held by James.
Added time: 1 minute.
45 - GOAL! Hammers take the lead at a vital stage of the match and Sinclair makes up for his earlier miss as he collects a Lomas cross in the area with his back to goal and turns Moore before firing in a superb left-foot shot from an acute angle beyond the dive of Hoult. West Bromwich Albion 0 West Ham United 1.
Half-time: West Bromwich Albion 0 West Ham United 1.
Second Half:
46 - Hammers have a decent shout for a penalty turned down, as Di Canio appears to be tugged in the area by Moore, but referee Dean waves away the appeals.
48 - Substitution for West Ham, Defoe replaces Di Canio.
50 - GOAL. Disaster for Hammers, as the hosts equalise with their first attack of the half. Koumas' corner is cleared by Ferdinand, but the ball is pushed back out wide, where Koumas sends in a cross that is met by the head of Dichio. James gets a hand to the effort but can't prevent it from finding the net. West Bromwich Albion 1 West Ham United 1.
51 - Carrick booked for dissent.
52 - With West Brom now revitalised, a second goal seems likely as Hughes collects in the area and volleys goalwards from eight yards out, but James pulls off a stunning reflex save to keep the ball out.
57 - After Brevett is penalised for a foul right on the edge of the area, Koumas' free-kick deflects off the wall and lands in the six-yard box, where Carrick clears in the nick of time as Hughes gets ready to pounce.
67 - GOAL! Against the run of play, Hammers regain the advantage thanks to Sinclair's second of the afternoon. Bowyer's cross from the right is volleyed back into the danger area by Defoe for Ferdinand, whose header rebounds off the bar and is nodded home by Sinclair. West Bromwich Albion 1 West Ham United 2.
69 - Sinclair almost turns from hero to villain in an instant, as his weak header back to James falls perfectly for Hughes, whose volley is destined for the top corner until the Hammers' keeper amazingly tips it round the post for a corner.
74 - Double substitution for West Brom, Dobie replaces McInnes and Sigurdsson replaces Wallwork.
75 - Dobie has an immediate chance to level the scoring, as he bursts clean through but fires his shot straight at James from the edge of the penalty area.
76 - James races out of his area to head clear a loose ball but only succeeds in finding Dichio, who heads the ball straight back. Thankfully, it drifts wide.
78 - Bowyer collects from Defoe on the halfway line and strides forward but, instead of releasing the well-placed Ferdinand, fires in a low shot that is held by Hoult.
80 - The hosts again breach the Hammers defence as Hughes this time breaks clear, but once again James is in unbeatable form and blocks the effort before Pearce clears the loose ball.
81 - Substitution for West Ham, Hutchison replaces Ferdinand.
85 - Substitution for West Ham, Breen replaces Bowyer.
Added time: 3 minutes.
Full-time: West Bromwich Albion 1 West Ham United 2.