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Manchester United’s Altay Bayindir celebrates after he saves a penalty from Arsenal’s Kai Havertz during the shoot-out.
But, after a brief melee which saw Maguire, Havertz and Gabriel booked, Bayindir produced a superb save from Arsenal captain Martin Odegaard in the 72nd minute.
The Gunners sought to press their numerical advantage as United sat deeper and deeper, with substitute Declan Rice heading straight at Bayindir when unmarked.
Havertz then spurned a golden chance late on, slicing horribly wide from close range, and Bayindir again saved from Rice at the death to ensure another 30 minutes.
De Ligt produced two moments of brilliance in extra time, blocking a Rice shot and then preventing Leandro Trossard from pulling the trigger from two yards out as United clung on.
United almost nicked an unlikely winner in the second period of extra time, with David Raya just pushing Zirkzee’s goalbound shot wide.
But defending champions United, who needed penalties in last year’s semi-final against Coventry City en route to their 14th FA Cup triumph, prevailed in the shootout.
Despite a difficult start for new boss Ruben Amorim, who suffered his first defeat as United coach at Arsenal in the league last month, Sunday’s victory suggests United may have turned a corner.
“Especially in the first half, we played well, better than in the first game,” Amorim told reporters.
They will play Leicester City, managed by former United great Ruud van Nistelrooy, at home in the fourth round.
Arsenal boss Mikel Arteta, meanwhile, was left to rue his side’s inability to score.
“We didn’t get what we deserved,” he told reporters.