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England international Welbeck, who joined the Londoners in September after his United career stalled, pounced on a horrendous back pass by Antonio Valencia in the 61st minute to send the holders through.
In a rip-roaring first half Nacho Monreal's cool finish put Arsenal ahead in the 26th minute but Wayne Rooney's diving header evened it up three minutes later.
United, who will now ponder a second consecutive season without silverware, ended the match with 10 men after Angel Di Maria was red-carded for grabbing the referee's shirt just seconds after being booked for diving.
The goals have dried up for Welbeck since his early burst of form at Arsenal but, on his first appearance at Old Trafford since his 16 million pounds ($24.20 million) move, he produced the game's decisive moment with a helping hand from Valencia.
There seemed little danger as a high ball was pumped forward but Phil Jones elected to chest the ball to Valencia who wafted a weak pass in the direction of keeper David De Gea.
It never stood a chance and Welbeck clipped the ball round the Spaniard and rolled it into an empty net for only his third Arsenal goal since October.
With a Wembley semi-final against second tier Reading or third tier Bradford City to come next month, Arsenal are favourites to retain the trophy and claim a record 12th FA Cup.
The other semi-final will be between Aston Villa and Liverpool or Blackburn Rovers.
A lively start to the game saw Marouane Fellaini squander a chance for United before Alex Oxlade-Chamberlian carved open the home defence with a mazy dribble to tee up Monreal.
With fellow countryman De Gea to beat, the left back poked an effort inside the near post.
The lead lasted three minutes with Rooney, who has still not won the FA Cup, heading in Di Maria's pinpoint cross.
United struggled after the break and Arsenal could have won by a bigger margin with Alexis Sanchez shooting just wide late on.