Spurs led 2-0 at half-time only for West Ham to recover to win 3-2 and join Chelsea, 2-1 winners over Everton, in the last eight. Moussa Sissoko and Dele Alli put Spurs in control at Wembley but Andrew Ayew scored twice in five minutes to level the match after an hour.
Angelo Ogbonna then won it for West Ham with the third goal 20 minutes from time, with a superb header into the corner.
Antonio Ruediger put Chelsea in front after 26 minutes and Willian doubled their lead on 90 minutes.
Dominic Calvert-Lewin gave Everton a glimmer of hope with a goal seconds later but Chelsea hung on for victory.
