The England international marked his first return to his former club since leaving for north London during the close-season by capitalising on a weak Antonio Valencia backpass in the 61st minute and sending holders Arsenal into the last four.
Nacho Monreal had given the visitors the lead with just his second goal for the club in the 25th minute, but Wayne Rooney restored parity by heading in from a superb Angel di Maria cross soon after.
Welbeck had the final say, though, as he rounded David de Gea to roll into an empty net for Arsenal's first win at United since 2006.
Di Maria saw red late on for a tug on referee Michael Oliver's shirt immediately after being booked, and United's focus now switches to chasing down a top-four finish in the Premier League.
Welbeck was chosen ahead of Olivier Giroud in the Arsenal XI, while Luke Shaw came into the home defence as Marcos Rojo shifted into the centre to cover for the absent Jonny Evans, suspended for six matches over the weekend for spitting at Newcastle United striker Papiss Cisse last week.
Alexis Sanchez probed the United back line as Arsenal started brightly, but it was United who began to look the more likely to break the deadlock – Marouane Fellaini's hesitance in the penalty area ending one promising situation after 14 minutes.
Arsenal withstood that pressure, though, and eventually went ahead in the 25th minute – an unmarked Monreal slotting home from close range following some neat footwork by Alex Oxlade-Chamberlain on the edge of the box.
But that lead lasted just four minutes, as Rooney got his head to a delightful Di Maria cross from the right for his 14th career goal against Arsenal.
Welbeck had appeals for a penalty turned down when he felt he was held back by Rojo as he ran into the box, while Wojciech Szczesny had to be alert to keep out a low Di Maria strike at the other end.
Louis van Gaal saw the need to make two half-time changes, bringing Michael Carrick and Phil Jones on for Ander Herrera and Shaw, before injury forced Oxlade-Chamberlain off to be replaced by Aaron Ramsey.
The England international pulled up after challenging Rojo for the ball.
Lacking the rhythm of the opening 45 minutes, goalscoring opportunities were at a premium at the beginning of the second half, but Di Maria dragged an effort wide on the hour.
However, a Valencia slip-up at the other end cost United dear, as Welbeck pounced to punish the Ecuadorian full-back with a composed finish.
There was no time for sentiment, though, as the striker celebrated a goal against his boyhood club with passion.
Hector Bellerin could count himself lucky to stay on the pitch when he risked earning a second yellow card with a clumsy challenge on Ashley Young, and Arsene Wenger soon took evasive measures by replacing the full-back with Calum Chambers.
Chris Smalling lifted over from close range with United's best chance to pull level and the home side's frustrations were clear when Welbeck left the field to heckles from some sections of the ground.
De Gea got down brilliantly to keep out Santi Cazorla soon after, but United's chances looked over when Di Maria was dismissed in bizarre circumstances.
Booked for diving, the Argentine pulled the shirt of referee Oliver to earn a second yellow and his marching orders.
Substitute Adnan Januzaj then went down in the box under the challenge of Monreal with three minutes left on the clock, but he was rightly booked for simulation as Arsenal held on for the win.
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