Former US Open champion and five-time Grand Slam runner-up Medvedev was dumped out in the first round for the sixth time in Paris, losing 7-5, 6-3, 4-6, 1-6, 7-5.
British number one Norrie, ranked 81, took just under four hours to wrap up his first win in five meetings against the 11th-seeded Russian.
"It was a great atmosphere, it really felt I had momentum in the first set and he raised his level," said Norrie.
"I managed to get it over the line, it was a crazy match, he's got me the last four our five matches."
The 29-year-old Medvedev's best finish in Paris was the quarter-finals in 2021.
Norrie, also 29, plays the second round in Paris for the fifth time where he meets either American Aleksandar Kovacevic or Argentina's Federico Gomez.