Grant Ward's deflected free-kick was cancelled out by Ravel Morrison's volley in the first half.
A back-pass was caught by the wind with Nathan Oduwa setting up Harry Winks to put us 2-1 up 12 minutes after the break, before Ward completed the scoring in the fifth minute of stoppage time with a carbon copy of his earlier free-kick.
West Ham: Spiegel, Pike, Page, Nasha, Burke, Oxford, Parfitt-Williams (Turgott, 76), Whitehead, Lee (C) (Lletget, 46), Morrison, Sadlier. Unused subs: Makasi, Howes (GK), Potts.
Spurs: Archer; Yedlin (McQueen, 60), Ball, Khumalo (C), Ogilvie; Ward, Winks (Lesniak, 69), Onomah; Miller (Sonupe, 70), Akindayini, Oduwa. Unused subs: Miles (GK), McEvoy.