Diafra Sakho scored within two minutes at Old Trafford when these sides met in the Premier League at the weekend and Zlatan Ibrahimovic repeated the trick here, lifting the ball over the keeper from Mkhitaryan's backheel.
Ashley Fletcher scored against his former side to leave the sides level at half-time, but a brace from Anthony Martial in the second period saw the home side through at ease.
Early opener from Ibrahimovic
The opening goal of the game came within two minutes of the kick-off, as Ibrahimovic and Henrikh Mkhitaryan inexplicably found themselves with time and space at the edge of the area on the left-hand side of play.
The winger spotted Ibrahimovic's run in behind him and sent him clear on goal with a delicate backheel, with the Swede making no mistake with a simple low finish into the bottom left.
Five minutes later, it could have been two as the home forwards again found themselves with the freedom of the West Ham defensive third.
This time, Ibrahimovic had options left and right but chose to drive forward and go for goal himself, but Adrian was equal to his shot and the defence eventually managed to hack the ball clear.
United make chances but Hammers draw level
Wayne Rooney tested the keeper with a dipping free kick from the corner of the area, Martial blazed well over the bar at the end of a counter attack and Ander Herrera saw a powerful drive blocked as United made the better chances in a half which saw neither side kick into top gear.
However, the Hammers were level with ten minutes of the half remaining. Dimitri Payet drove in from the left and let fly with a swerving low shot which David de Gea could not hold, with Fletcher perfectly placed to bundle home the rebound.
The first half somewhat fizzled out as a spectacle with little more to note aside from the injury-enforced withdrawal of Aaron Cresswell through injury, and the two sides went in level at the break.
Martial brace puts hosts in control
Despite this sluggish end to the first half, the home side started the second as rapidly as they had the first and soon regained the lead. Mkhitaryan was at the centre of things again, drifting in from the right this time and pulling the ball back for Martial to finish with aplomb at the back post for his third goal of the season.
He almost had his fourth ten minutes later as he was set free by Rooney's incisive pass, but he fired straight at the onrushing Adrian from just inside the penalty area.
Soon after, West Ham's injury problems began to pile up as Michail Antonio followed Cresswell down the tunnel after sustaining a knock inside his own half, and was subsequently replaced by Manuel Lanzini.
Martial did not have to wait too long for a second goal of the night. With an hour played, Antonio Valencia found Ibrahimovic in space down the right and he swept the ball through the centre of the penalty area to give the France international a simple one-on-one finish.
United dominate under-par visitors
Despite the lack of a managerial presence on the touchline in the wake of Jose Mourinho's recent suspension, Manchester United continued to enjoy the bulk of possession and chances.
Ibrahimovic attempted the spectacular with an overhead-kick cross towards Rooney, but Winston Reid hooked the ball away before Rooney could connect. His boot drew blood from the face of the England captain - with both already booked - drawing a furious reaction from Rooney at the referee's refusal to send the West Ham skipper off.
United went close again through Rooney before Simone Zaza's disappointing loan spell from Juventus continued with a wayward shot that spun out of play for a throw-in further back than the Italy forward had taken it.
Perhaps the biggest cheer of the night accompanied the introduction from the bench of Bastian Schweinsteiger in the closing minutes, as the Old Trafford fans gave a clear show of support to the former Germany international with every subsequent touch of the ball loudly cheered.
Ibrahimovic rounded off the night to complete his brace from Herrera's cut-back in the last minute of stoppage time, and his side will now go on to face one of Liverpool, Hull City or Southampton in the final four.