Man United made six changes from the draw with Everton, as goalkeper Dean Henderson, Anthony Martial, Fred, Donny van de Beek, Nemanja Matic and Alex Telles all got a run in the line-up.
After a quiet start, Victor Lindelof came the closest to breaking the deadlock in the 27th minute with a deflected header that was tipped onto the post by shot-stopper Lukasz Fabianski.
West Ham were growing in the game in the closing phase of the first stanza but they could not carve out clear chances, only threatening on set-pieces while Ole Gunnar Solskjaer's charges lacked creativity in the final third and the first stanza.
Marcus Rashford came close to giving United a great start eight minutes from the restart after he latched onto a rebound. but his close-range half-volley was saved by Fabianski.
With the home side struggling to carve out chances, Solskjaer made a double change in the last quarter of an hour after bringing on Bruno Fernandes and Scott McTominay in place of Van de Beek and Matic.
Martial almost snatched the winner for Man United in the last eight minutes with a powerful header at the near post that missed just wide, sending the clash to extra-time.
McTominay finally broke the deadlock for United in the eighth minute of extra-time with a first-time strike into the bottom corner after he was teed up by Rashford.
Mohamed Benrahma had a chance to get the equaliser for West Ham with six minutes left after receiving a cross unarked inside the six-yard box but his free header was saved by Henderson, as the Red Devils marched on to the last eight.
