Everton caretaker manager David Unsworth's first game on the bench ended in a 2-1 defeat by Chelsea at Stamford Bridge.
Spurs looked comfortably in charge against West Ham after Moussa Sissoko put them ahead in the sixth minute with a side-foot finish after being set up by South Korean winger Son Heung-min
Slaven Bilic's side fell 2-0 behind in the 37th minute when Son set up Dele Alli whose shot deflected in off Hammers' defender Declan Rice.
Tottenham, who rested top scorer Harry Kane, were strolling towards the last eight until Andre Ayew brought West Ham back into the game in the 55th minute with a tap-in after Edimilson Fernandes's strike was pushed out by keeper Michel Vorm.
Five minutes later Ayew equalised, slotting home after Andy Carroll had headed into the path of Manuel Lanzini.
The comeback was complete in the 70th minute when defender Angelo Ogbonna met a Lanzini corner at the near post with a glancing header.
"We deserved it and of course we are delighted and it is a big boost for us," said Bilic whose future has been the subject of speculation.
Antonio Rudiger's 26th-minute header and a late strike from Willian secured Chelsea's progress into the last eight.
Everton, playing their first game since the sacking of Ronald Koeman, delivered an improved performance but it was not enough to stop the London club.
From a short corner, Charly Musonda swung in a deep cross and Rudiger rose well to power a header past Everton keeper Jordan Pickford.
Both teams made several changes from the line-ups that featured in the Premier League last weekend and while Everton enjoyed good pressure in the second half they were unable to find a way through.
A low shot from Willian which flew in off the post in stoppage-time doubled the advantage for Antonio Conte's team, although Everton forward Dominic Calvert-Lewin swiftly responded with a good finish to give the Merseyside club a consolation.
The quarter-final draw takes place on Thursday with holders Manchester United, Premier League leaders Manchester City, Arsenal, Bournemouth, Leicester City and Championship side Bristol City also in the hat.
(Reporting by Simon Evans, editing by Ed Osmond)
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