Marcus Rashford gave the Reds the lead in this quarter-final replay on 54 minutes, the 18-year-old collecting Anthony Martial's pass and curling a magnificent finish past helpless Hammers keeper Darren Randolph.
When Marouane Fellaini- who had his best game of the season- bundled home a Martial cross 13 minutes later it looked all over- only for United to come under siege in the closing stages.
United keeper David de Gea- in another virtuoso display- had already saved superbly from Michail Antonio when James Tomkins stooped bravely among flying footwear to halve the deficit 11 minutes from time.
The hosts finally hammered at United's rearguard in a belated spell of pressure with De Gea saving once more from Andy Carroll and Cheikhou Kouyate, the latter flagged narrowly offside as he headed home a rebound.
Enner Valencia went close for West Ham early on whilst Randolph tipped over Fellaini's flick and then saved well from Jesse Lingard after fine build up from Rashford.
The Reds were playing with a verve, pace and energy conspicuous by their absence this season and- after Lingard fired narrowly wide- took the lead within ten minutes of the resumption.
Rashford picked up Martial's pass, shimmied and curled a brilliant effort into the top corner for his sixth goal in a whirlwind United career.
The hosts fired a couple of speculative efforts at De Gea as they pushed for a leveller, but it was the Reds whom deservedly doubled their advantage on the hour mark.
Lingard crossed for Martial who controlled and fired across goal but the ball deflected off Tomkins and was diverted in off the knee of Fellaini and put United within sight of Wembley.
West Ham did not go quietly, however and roared back at their visitors as a slow burning cup tie turned into a fully grown thriller.
After an astonishing point blank save from Antonio, the Hammers halved the deficit to set up a frantic finale.
Carroll- subdued throughout after his seven-minute hat-trick on Saturday, nodded back across goal from a Mark Noble corner and Tomkins untidily forced the ball home from five yards.
That goal spurred Slaven Bilic's side into life as the hosts laid siege to the United goal.
De Gea prevented Chris Smalling putting through his own net, Carroll headed over when well-placed and the Spanish keeper made a superb late double save to keep out first Kouyate and then Carroll.
Kouyate's header was ruled out for offside, Wayne Rooney made his return from injury and Manuel Lanzini lashed wide but United held out to book a semi-final against Everton on 23rd April.
Overall team performance: 8.5/10
United Faithful Man of the match: Marcus Rashford (De Gea and Fosu-Mensah were also excellent)