A strike from Mitchell Cole wasn't enough to see off the Gunners, who are now favourites to progress.
The Hammers started the better side and dominated proceedings in the first half, but wasted three good chances to take the lead and were punished for their errors midway through the second half when the Gunners took the lead.
The deficit spurred the Hammers to fight back and a deserved equaliser arrived 10 minutes later, when the ball ran loose to the edge of the area and Cole blasted a stunning shot into the top corner.
That gave Peter Brabrook's side extra determination, but with just eight minutes remaining, Arsenal left their opponents shell-shocked as they grabbed the clinching goal.
A third was added in the dying moments as West Ham threw everyone forward in search of another equaliser and the final whistle soon after marked the end of Hammers' participation in this season's competition.
Said Peter Brabrook afterwards: "It's very disappointing to have lost the game, because we were by far the better side in the first half and should have been two or three goals up at half-time.
"Once again, it was the problem that has plagued us all season - not being able to take our chances, and we were punished for it in the end.
"There was also some bad defending for their first two goals and we have contributed to our own downfall really.
"I can't fault the effort of the players, they have given everything but, as they go higher up the ladder, they will have to improve their defending and take their chances if they want to progress further.
"That is the season over for us now, but a lot of the first year boys will now play for the under-19s until their season finishes, and we also have a tour to Hong Kong at the end of the season that will be a good experience for the boys."
Team: Sealey, Lumsden, Ward, Ferdinand, Tattam, Akinsete, Carrick, Saunders (Bunce), Cole (Fletcher), Pearson (Collington), Morris. Subs unused: Read.
Meanwhile, a makeshift under-19 side battled bravely to secure a 1-1 draw with Chelsea on Saturday morning.
Despite having several players out of position, five schoolboys in the starting line-up and NO substitutes at all, a team led by under-16 coach Paul Heffer grabbed a point thanks to Freddy Eastwood's second half strike.
Team: Jackson, Tucker, Blewitt, McClenahan, Metitiri, Laws, Mehmet, Parrington, Lee, Eastwood, Khan.