The 18-year-old from Sheffield, England finished with a one-over 72 for a 10-over total for his four rounds over a Muirfield course that has separated the men from the boys this week.
After practising with American great Tom Watson on Wednesday, Fitzpatrick signed off after a round with former Masters winner Fred Couples who finished one shot better.
"I couldn't be happier with the way I played out there. It's been a really good day," said Fitzpatrick, who looks so young that he has regularly been asked to show his ID this week, told reporters.
"I will miss it. It's back to reality."
Fitzpatrick, who is about to begin a golf scholarship at Northwestern University in Chicago where former world number one and fellow Briton Luke Donald studied, impressed Couples.
"He's very good. He doesn't miss many shots, hit it straight. As he gets bigger, he'll hit it a little further, but he hits it plenty far," said the American veteran who turned professional 14 years before Fitzpatrick was born.
"He's not like one of these crazy bombers that you see. He's good. At 18 you should putt well. He's a good putter."
By making the cut at Muirfield, Fitzpatrick did far better in his first Open than Donald and world number Rory McIlroy, who both missed the cut, and he finished in front of twice former Open champion Padraig Harrington.
(Reporting by Martyn Herman, editing by Ed Osmond)
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