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.
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.
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.
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.
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.
