The other semi-final sees Aston Villa take on the winner of Blackburn and Liverpool's replay after the pair drew 0-0 at Anfield on Sunday.
Arsenal's Danny Welbeck scored on his return to Old Trafford to seal a 2-1 victory for the 2014 FA Cup winners after Wayne Rooney had cancelled out Nacho Monreal's opener.
That has set up a clash with Sky Bet League 1 Bradford, who knocked the Gunners out of the Capital One Cup in 2012, or Steve Clarke's Reading.
Villa secured their place in the semi-finals for the first time since 2010 by seeing off local rivals West Brom 2-0 in a fiery affair on Saturday and Tim Sherwood's side will now be waiting to see whether Liverpool or Blackburn are their Wembley opponents for a place in this season's final.
The semi-final ties will be played across the weekend of April 18 and 19, meaning Arsenal's Premier League game against Sunderland will be rearranged, as will Villa's home game against QPR which is currently scheduled for Monday April 20th.
The FA Cup
Man Utd 1 - 2 Arsenal
Liverpool 0 - 0 Blackburn
A Villa 2 - 0 W Brom
Bradford 0 - 0 Reading
After seeing off Louis van Gaal's Red Devils in the quarter-finals, Wenger insists his Arsenal side cannot rest on their laurels and believe a place in the final is already in the bag.
"Last year we played a tricky game at Wembley in the semi-final and we expect that again," he told BBC1.
"Tonight I thought we played well and deserved to win the game. We live in the real world and want to compete in every competition. This result can give us good morale for the Premier League and the game in Monaco (in the Champions League)."