The England striker was surplus to requirements at Louis van Gaal's United, but was quick to capitalize on a mistake by Antonio Valencia to grab the winner just after the hour mark.
A disappointing night for United was compounded by a late red card for Argentina international Angel Di Maria, who was given his marching orders after manhandling referee Michael Oliver.
Di Maria was initially booked for simulation and took his protests too far by grabbing the official, which led to his instant dismissal.
With 10 men, United was never able to exert any real pressure on Arsenal and it took two fine saves David De Gea to deny further goals from Santi Cazorla and Alexis Sanchez for the away side.
Victory for Arsenal booked a return to Wembley for a last four clash with either Reading or Bradford City, who are replaying their quarterfinal tie next Monday.
Liverpool or Blackburn Rovers will play Aston Villa in the other semifinal.
Arsenal had taken a 25th minute lead through Nacho Monreal, the Spanish defender converting after fine move which involved Mesut Ozil and Alex Oxlade-Chamberlain.
But the advantage was short lived with Di Maria finding Wayne Rooney to score with a diving header.
United looked to press ahead for the winner, but home hopes were dashed when Valencia made a hash of a back pass and Welbeck was quick to latch on to the loose ball and beat De Gea.
'I believe it was justice for him to score he worked extremely hard today,' Arsenal manager Arsene Wenger said of Welbeck.
His counterpart van Gaal will now, barring a miracle, be left without a trophy in his first season in charge of the English powerhouses, with the pressure now on to seal a top four place in the EPL to guarantee Champions League football next season.