Roman Michael-Percil, Shaq Coulthirst and Kenny McEvoy scored in the space of 10 minutes to put the lads 3-0 up and looking set to run riot at Upton Park.
But a moment's lapse in concentration allowed the Hammers back into the game before the break and a second goal in the 57th minute gave the home side real hope of an unlikely comeback.
The sheer pace of McEvoy, Michael-Percil and Coulthirst meant we were always a threat going forward, but the key fourth goal came in unlikely circumstances as Grant Ward's speculative 35-yarder found the top corner.
McEvoy's second and our fifth late on gave the scoreline a more of a reflection of the 90 minutes.
The lads looked in complete control midway through the first half.
The first goal arrived on 11 minutes as McEvoy timed his cross perfectly for Michael-Percil sliding in at the far post.
Michael-Percil was involved in the second on 17 minutes, drilling in a cross from the right that Coulthirst met with a lovely angled volley into the corner.
McEvoy scored the third on 22 minutes, latching onto Luke McGee's clearance and racing into the box before calmly sliding the ball past Gines Rosqiue.
West Ham were offered a flicker of light in the 37th minute when Laste Dombaxe miscontrolled on half-way, allowing Elliot Lee to charge into the box unchallenged and he finished well across McGee.
Dombaxe and McEvoy had early efforts in the second half but there were signs that West Ham were starting to believe a little more, led by Kieran Bywater and Lee.
Bywater let rip from 25 yards, forcing a fine tip over from McGee but it was soon 3-2 as we failed to clear our lines and Lee crossed for Bywater to head home at the far post.
Coulthirst was somehow denied twice from point-blank range seconds later and we remained lively in the attacking third and always likey to score, McEvoy fashioned an opening on 67 minutes and drilled straight at Rosqiue.
Despitre all that fine attacking play, the fourth goal, key on the night, came from nowhere as Ward curled towards goal from fully 35 yards, Rosqiue misread the ball's flight and it ended up in the top corner.
That goal burst the home team's bubble and it was no surprise when McEvoy added the fifth in the 84th minute, tapping in after Ward won possession and Coulthirst turned Emmanuel Onariase before forcing a fine save from Rosqiue.
West Ham (4-1-4-1): Rosqiue; Alexander, Harney, Nasha (Marlow, 77), Shaw; Miles (Onariase, 70); Mavila (Gogo, 82), Cullen, Bywater, Sadlier; Lee. Unused subs: Brown, Page.
Spurs (4-3-3): McGee; McQueen, Ogilvie, Veljkovic, Ward; Dombaxe (Lesniak, 85), Bentaleb, Onomah (Winks, 50); Michael-Percil (Akindayini, 81), Coulthirst, McEvoy. Unused subs: Ball, Priestley.
Goals: West Ham - Lee (37), Bywater (57). Spurs - Michael-Percil (11), Coulthirst (17), McEvoy (22, 84), Ward (75).