Angel Di Maria's 77th-minute sending-off for two yellow cards — one for diving and the other moments later for tugging the referee's shirt in protest — completed a disappointing evening for United, which is almost sure to go without a trophy for the second straight season.
United sold Welbeck on transfer deadline day last summer, with manager Louis van Gaal saying he wasn't up to the standard of the club's strikers, but he showed great instincts to pounce on Antonio Valencia's poor back-pass and score in the 61st.
"You always want to do well when you play against your former club, to show you are a great player," said Arsenal manager Arsene Wenger, who added that he chose to start Welbeck for "psychological considerations."
Wayne Rooney's diving header in the 29th cancelled out Nacho Monreal's well-worked opener four minutes earlier for the visitors, who will play third-tier Bradford or second-tier Reading in the semifinals at Wembley Stadium. Aston Villa was drawn against the winner of the replay between Liverpool and Blackburn.
It was a rare start up front for Welbeck, who has had a mixed first season for Arsenal since his move for 16 million pounds (then $26.5 million) and is the second-choice behind Olivier Giroud. And he took his only real chance of the game well after Valencia underhit his back-pass, pouncing on the ball to round David De Gea and slot into an empty net.
He celebrated in front of the Arsenal fans, clenching his fists. Welbeck, who started at United at the age of 8, was applauded by most of the home fans when he was substituted in the 74th.
"It is very disappointing when you give the victory (away)," Van Gaal said. "You don't lose from your opponent, you have lost from yourself."
The FA Cup's two most successful teams — with 11 titles apiece — were taking a break from the race for a top-four finish in the Premier League, which Van Gaal says is his priority in his first season at Old Trafford. Arsenal is third while United is fourth, 10 points behind leader Chelsea with 10 games left.
Arsenal carved United apart for Monreal's goal. Alexis Sanchez set Mesut Ozil free and the ball was passed across to Alex Oxlade-Chamberlain. The winger jinked past three defenders and drew in a fourth before finding Monreal, whose shot curled inside the near post.
United's response was immediate, as Di Maria cut inside — a familiar tactic in the first half — and whipped in a cross that Rooney met with a diving header. It was Rooney's fourth in five games since returning to his favoured role as central striker.
United finished the half stronger, with Wojciech Szczesny — recalled as Arsenal's goalkeeper — saving from Di Maria and Fellaini, but Arsenal always looked threatening on the counterattack with Welbeck as a pacy lone striker.
And he showed great opportunism to pounce on Valencia's back-pass, nip the ball to the side of the outrushing De Gea and steer it into an empty net. Santi Cazorla and Sanchez had efforts saved by De Gea in the closing stages as Arsenal threatened more goals.
United rarely threatened after Di Maria's dismissal, which came after he was booked for tumbling under a challenge from Aaron Ramsey. He got up to protest and grabbed referee Michael Oliver, who produced a second yellow card.
"I know of course and also Angel Di Maria knows that he doesn't have to touch the referee," Van Gaal said, "so that is not so smart of him. But to control emotion is not so easy in such a match."