With penalties looming, the United captain broke the Canaries’ resistance with a drilled close-range effort with two minutes remaining at Carrow Road.
Odion Ighalo had given United the lead early in the second half but Todd Cantwell’s 20-yard drive, and an inspired performance from goalkeeper Tim Krul, forced extra-time after Timm Klose’s dismissal for a last-man challenge on Ighalo.
Krul thwarted United time and time again but the visitors found the winner with seconds remaining to book a semi-final spot for Ole Gunnar Solskjaer’s side for the third time in six seasons.
Given there was a semi-final spot at stake there was precious little quality in a dull first half at Carrow Road, with Norwich starting cautiously against a side that had put seven goals past them in their two league meetings this season, and United disjointed after eight pre-match changes.
Solskjaer resisted introducing substitutes at the break and his faith in his starting XI was repaid within six minutes of the restart. Shaw’s drilled cross was sent looping into the air by the Norwich defence before falling for Ighalo, who showed great composure to volley United into the lead.
It was Ighalo’s fourth goal in four starts for the club, but United were guilty of taking their foot off the gas as Norwich steadily grew back into contention.
The pressure finally told though, with Cantwell capping an impressive individual performance with a rasping drive into the bottom corner from 25 yards that Sergio Romero, perhaps unsighted by the sun, should have saved.
Norwich kicked on for the winner as substitute Josip Drmic dragged a shot inches wide within minutes of coming on, but it was backs to the wall for the Canaries with two minutes remaining after Klose was shown a straight red card for denying Ighalo a goalscoring opportunity.
Tim Krul thwarted substitute Paul Pogba’s attempt to score the winner in the last of six added minutes, as the 10 men forced extra-time for the second tie running.
Pogba glanced a header wide with seconds of the restart and Marcus Rashford drew a fingertip save from Krul with a dipping drive from 18 yards, before Pogba ended the first period of extra-time with a thunderous drive that Krul spilled before gathering at the second attempt.
Krul clawed Maguire’s goal-bound header off the line and tipped Bruno Fernandes’ dipping strike but was beaten with two minutes remaining as Pogba’s delivery in from the left caused panic in the Norwich defence and the ball eventually dropped to Maguire to convert from six yards.