Arsenal's Kai Havertz was the only one of the first four penalty takers to miss, denied by a brilliant stop from Altay Bayindir, before Zirkzee – booed off in a defeat by Newcastle United last month – stepped up to win the shootout 5-3.
"I just want to help this great team," second-choice goalkeeper Bayindir told the BBC. "I want to make everyone happy for this great club ... if you are not playing it doesn't matter.
"You have to be ready every minute, every second – if you are a Man Utd player you have to be ready always."
After an underwhelming first half, United took the lead in the 52nd minute through Bruno Fernandes' curling effort from just inside the box after a great run by Alejandro Garnacho.
The game turned on its head 10 minutes later though, as United's Diogo Dalot – already booked for a wild challenge on Myles Lewis-Skelly – received a second yellow card for a reckless tackle on Mikel Merino.
Two minutes later, Bayindir's poor attempted punch fell at the feet of Gabriel, whose shot took a wicked deflection off Matthijs de Ligt and in to put the Gunners in charge of the tie.
Arsenal could have completed the comeback when they were awarded a questionable penalty after Havertz went down under minimal contact from Harry Maguire.
But Bayindir produced a superb save from Arsenal captain Martin Odegaard, before the Gunners spurned several chances to score a winner.
United almost nicked an unlikely winner themselves 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.