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 a trophy in the wake of manager Alex Ferguson's retirement, 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.
Welbeck had managed only two goals for Arsenal since October and was not having much joy on his first return to Old Trafford since his 16 million pounds (24.20 million) switch.
But he produced the game's decisive moment with a helping hand from United's bungling defence.
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.
"We had to start without apprehension and play at a high pace and we did that well," said Arsenal manager Arsene Wenger, who admitted in the build-up that he was still haunted by a 1999 FA Cup semi-final defeat by United.
"Danny Welbeck, I believe is just happy to score. He worked extremely hard today and deserved his goal."
It was Arsenal's first FA Cup win over United since the 2005 final and their first in any competition since May 2011.
WEMBLEY DATE
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.
Liverpool will have other ideas, although they must win a replay against Blackburn Rovers to book a semi-final spot against Premier League rivals Aston Villa.
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 Santi Cazorla and Alexis Sanchez both denied by the superb De Gea.
(Reporting by Martyn Herman, editing by Tony Jimenez and Justin Palmer)
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