The win extends United's unbeaten run in all competitions to 14 matches and secures a 30th appearance in the last four of the FA Cup for the club.
"It wasn't a spectacle, we didn't threaten enough or create enough chances," said United manager Ole Gunnar Solskjaer. "But I am delighted we are into the last four and it was a good workout, we got some minutes for a lot of players. We are through and that is what counts in the Cup."
Solskjaer made eight changes from the team which beat Sheffield United 3-0 in the Premier League on Wednesday, with a number of squad players given a start.
Solskjaer threw on Paul Pogba and Nemanja Matic as United looked to avoid extra-time and the momentum shifted back to his side when Norwich defender Timm Klose was sent off for pulling down Ighalo on the edge of the box in the 89th minute.
The game looked to be heading for a penalty shoot-out after Norwich defended superbly against constant pressure from Ole Gunnar Solskjaer's side before Maguire's goal two minutes from the end.
Norwich goalkeeper Tim Krul was up to the test in extra time, producing several fine saves to keep United from scoring, but with penalties looking inevitable Maguire popped up at the top of the six-yard box to send his team into the semifinals