Substitute Scott McTominay scored the winner in the first half of extra time as the Red Devils booked their place in the quarter-finals.
Wonderkid Amad Diallo was named on the bench for the first-time since his big-money transfer from Atalanta last month, but did not see any game-time.
So which players stood out most?
✔️ GOOD PERFORMANCE
Scott McTominay: 7.8
The Scotsman came off the bench to replace Nemanja Matic after 73 minutes and proved to be Man Utd's ultimate hero on the night in extra time.
He timed his run from deep to perfection as United launched a lightning-fast counter-attack. Without breaking stride, he fired home clinically from inside the area to break West Ham's resistance.
McTominay now has an impressive six goals in all competitions for Man Utd this season, a more than decent return considering the position he plays in.
Alex Telles: 7.7
It has been a fascinating fight for the left-back shirt at Old Trafford this season between Alex Telles and Luke Shaw, and the Brazilian did his chances no harm on Tuesday.
He was United's most available attacking outlet throughout the match, bombing up and down the left flank. He delivered an impressive 10 crosses in the normal 90 minutes alone and had over 100 touches.
He was taken off as the game entered extra-time after really putting in the miles.
❌ BAD PERFORMANCE
Andriy Yarmolenko: 5.8
Yarmolenko is having another miserable season in East London and has scored only one goal in all competitions.
He made no impact once again versus Man Utd. He made no dribbles, created no chances and had just one attempted shot which was blocked. Worst of all, he only made five passes in his time on the pitch.
It came as no surprise when he was hooked by manager David Moyes after 54 minutes.
Donny van de Beek: 6.0
Van de Beek continues to struggle to justify his big transfer fee and failed to make the most of his opportunity against West Ham.
Given an opportunity from the start by Solskjaer, Van de Beek made little impact. He had no dribbles or crosses and made no chances as once again his lack of creativity was apparent.
He attempted only one shot that was blocked before being subbed on 73 minutes.
⭐ STAR MAN
The Brazilian was everywhere and never stopped for the whole 120 minutes. He was involved in everything Man Utd did, making a game-high 154 touches.
His passing was excellent as he boasted a 90 per cent pass completion rate - with 43 of these passes in the final third.
He was also as tenacious as ever, winning nine duels and all of his aerial challenges.