Zverev made the perfect start, breaking Cobolli's serve in an opening game which saw the Italian make five unforced errors, and the German then held comfortably to race into an early lead.
He dominated the remainder of the opening set, serving excellently and taking full advantage of his opponent's nerves on his way to winning it 6-1 in just 35 minutes.
Cobolli went off court to regroup for the second set and it did the trick, with the underdog steadying the ship with some comfortable holds of serve.
He then got his first chance to break Zverev at 3-3, and while the world number three saved the first break point by winning a stunning 23-shot rally, he then hit a loose forehand to fall behind.
With the momentum behind him, Cobolli served things out twice to take the second set 6-4 and level things up.
It was now Zverev's turn to regroup, and he did just that, forcing two break points while leading 2-1 in the third but failing to convert them.
Things then stayed on serve until Cobolli made huge errors while trailing 5-4, going from 30-0 up to 40-30 down. Another mistake on his forehand followed, giving Zverev the third set, and moving him to within one of a first Grand Slam title.
Once again, Cobolli responded perfectly to falling behind, with a double fault from Zverev and a perfect backhand from the Italian giving him a break serve of serve in the opening game of set number four.
More unforced errors then came from Cobolli's racket, which allowed his opponent to draw level at 3-3, but Zverev handed the lead straight back to his opponent with a poor game.
Both players were clearly feeling the nerves at this point, and Zverev capitalised on Cobolli's with two stunning winners to get back on serve before taking a 6-5 lead to move to within one game of victory.
The German was having physical issues at this point, and after receiving treatment, could do little to stop Cobolli from taking things to a tiebreak. In it, a double fault from Zverev and a sumptuous drop shot from the Italian gave him two set points, and while he missed the first with a massive error at the net, he produced an unstoppable forehand to convert the second and take things to a decider.
Both were now fighting their own bodies and minds with the greatest prize of their career just a set away, and Cobolli looked to be very much losing those battles as Zverev stormed into a 3-0 lead before saving break point with some remarkable defensive work.
He finally held his nerve after that to seal a 6-1, 4-6, 6-4, 6-7, 6-1 victory and become a Grand Slam champion.