Despite enjoying the majority of possession and spending most of the time in their opponents' half, Tony Carr's team just couldn't find a way past a dogged and defensive Everton side and it was no real surprise when the tie reached a stalemate after 90 minutes.
The Hammers' biggest threat in front of goal came from lively striker Freddy Eastwood, who fought bravely to breach the visitors' rearguard and almost grabbed a deserved winner in the 70th minute when his fierce low drive from the edge of the area crashed against the post and rebounded away to safety.
The Toffees had been denied by the woodwork themselves in the 19th minute, when their only real effort of any note during normal time saw Michael Symes strike the crossbar from close range.
However, the hard-working Everton striker got his reward six minutes into the first period of extra-time when he took advantage of a lapse in the Hammers defence and slammed the ball past Glenn Jackson from 12 yards out.
Four minutes later, Symes effectively ended the tie after more tired defending from the hosts allowed him to find space on the right and fire a low shot past the despairing dive of Jackson.
Boss Carr immediately threw caution to the wind and pushed talented young defender Glenn Johnson up front, and the England under-19 starlet provided a glimmer of hope with three minutes remaining in the second period, when he jinked his way past two opponents on the left and fired in a low cross that was turned into his own net by defender Robert Southern.
Sadly for the young Hammers side, it was too little, too late, and their hopes of following in the footsteps of the class of '99 were laid to rest for another year.
Team: Jackson, Khan, Tattum, Lumsden (Dean 60), Ferdinand, Akinsete, Cascione, Johnson, D.McMahon (Ritchie 100), Allen (Carrick 100), Eastwood. Subs unused: G.Sealey, Metitiri.