The Premiership's worst travellers were ripped apart as Hammers added three second half goals to the opener secured after just four minutes of this one-sided game at chilly Upton Park.
Sebastien Schemmel scored his first for the club when he tucked away Don Hutchison's deflected free-kick from inside the six-yard box in front of the Bobby Moore Stand.
Apart from a couple of hopeful efforts midway through the first half, when David James finger-tipped Zavagno's low effort just wide and was tested by a lob from Carbone, the West Ham keeper had little else to do.
In the second half, it was all West Ham, who launched wave after wave of attack. Derby were obviously not helped by the sending off of Carbone, who received his second yellow card on 53 minutes, but they were already a beaten side well before Paolo Di Canio, Sinclair and substitute Jermain Defoe put them to the sword.
Di Canio deserved his 74th minute goal, drilled left-footed inside the far corner of the net, for supplying many of the party tricks that helped to warm the 31,379 crowd, although Joe Cole - outstanding again - and Hutch also produced their share of customary flicks and twists to delight the fans and mesmerise Colin Todd's men.
Sinclair's stunner came in the 86th minute, just seconds after Derby keeper Poom had denied Michael Carrick. As Joe Cole's deep, right-wing cross dropped towards him, Sinclair swivelled acrobatically in mid-air to score with a flying scissor-kick volley that will take some beating as goal-of-the-month. Ironically, it was his first goal in a Hammers shirt since Boxing Day last year, with the result today our biggest victory since that 5-0 drubbing of Charlton Athletic exactly a year ago.
Like Carrick, Cole wasted a good chance to score when his follow-up effort from sub John Moncur's shot was blocked, but young Defoe has no such inhibitions in front of goal.
Racing on to Carrick's neat through ball, the England under-21 striker prodded home Hammers' fourth goal in the 89th minute.
Defoe had replaced Fredi Kanoute, restored to the side after a recurring hamstring injury, who missed a fine chance himself when he toe-poked his shot just wide.
But with Di Canio ruled out of Saturday's home game against Liverpool due to suspension, it made sense for Glenn Roeder to withdraw the Frenchman and send on the eager Defoe, whose fifth Premiership goal of the season leaves him just one adrift of top scorer Kanoute.
It will be interesting to see how these two perform together up front against the title-chasing Reds.
4 - GOAL! Hammers' first attack of the game brings a goal as James' long kick upfield finds Di Canio, who is brought down on the edge of the penalty area. From the free-kick, Carrick touches the ball to Hutchison, whose shot is deflected into the six yard box, where right-back Schemmel is surprisingly waiting to stab home the loose ball. West Ham United 1 Derby County 0.
10 - Kanoute pulls clear of his marker on the right and clips in a cross that Di Canio meets on the volley. Sadly, his effort is blocked by a defender.
11 - Hutchison wriggles into the penalty area and goes down under a challenge but referee Poll waves away claims for a spot kick.
16 - Cole's measured pass finds Di Canio on the left and the Italian turns Mawene before firing in a left foot shot that Poom pushes away.
18 - A hopeful ball over the top sets Powell away, but James rushes out to pluck the ball off the Derby skipper's head.
21 - Zavagno booked for not retreating 10 yards at a free-kick.
22 - Zavagno's stinging left foot shot is tipped superbly round the post by James.
26 - Sinclair's attempted cross is deflected towards Cole, whose first time volley from the edge of the area flies well wide.
45 - Kanoute latches on to Sinclair's flick and sees his shot hit the post but is adjudged to have fouled Higginbotham.
Added time: 1 minute.
Half-time: West Ham United 1 Derby County 0.
46 - Substitution for Derby County, Kinkladze replaces Ducrocq.
47 - Repka booked for ungentlemanly conduct.
49 - Cole collects the ball in the centre of the field, beats one opponent and then runs at the Derby defence before firing in a shot that Poom saves well.
50 - Carbone booked for diving in the penalty area.
54 - Carbone sent off for a second bookable offence after fouling Repka.
57 - Grenet booked for foul on Sinclair.
61 - Substitution for Derby County, Boulder replaces Riggott.
62 - Di Canio releases Kanoute with a superb first-time pass, but the French striker toe-pokes his effort just inches wide of the post under pressure from Boertien.
64 - Schemmel's surging run sets up Kanoute, who turns in the box and slides the ball across goal, but Boertien clears.
72 - Schemmel's speculative right foot shot flies just wide of the target.
74 - GOAL! Carrick's crossfield pass isn't dealt with by Grenet and Di Canio latches on to the ball before firing a low left foot shot past the despairing dive of Poom. West Ham United 2 Derby County 0.
74 - Substitution for West Ham United, Defoe replaces Kanoute.
81 - Hutchison's long range shot is deflected into the arms of Poom, before Cole then gains possession from the clearance and skips into the box before firing a stinging effort into the midriff of the Estonian keeper.
86 - A great move sees Di Canio's flick release Carrick. The midfielder then feeds Defoe, who is blocked off before the ball rebounds to Carrick again, who sees his effort parried by Poom for a corner.
86 - GOAL! Di Canio's short corner finds Cole, who tees the ball up and sends over a cross to the far post where Sinclair is waiting to acrobatically volley past Poom from eight yards out for his first goal of the season. West Ham United 3 Derby County 0.
90 - Substitution for West Ham, Moncur replaces Di Canio.
Added time: 2 minutes.
90 - GOAL! Carrick's through ball finds Defoe, who shrugs off a defender before poking a left-foot shot past Poom to round the day off for the Hammers. West Ham United 4 Derby County 0.
Full-time: West Ham United 4 Derby County 0.