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The second seeds came from a set down to beat top seeds Stephane Houdet and Nicolas Peifer 4-6 6-1 7-6 (8-6).
The Britons twice failed to serve out the match before eventually securing victory in two hours and 33 minutes after a nervy third-set tie-break.
"We didn't like to do it the easy way. It's a really special moment," Scot Reid, 24, said.
Reid will compete in the men's wheelchair singles final on Sunday.
The Australian Open singles champion will play Sweden's Stefan Olsson, the number seven seed, for the chance to become Wimbledon's first men's wheelchair singles champion.
It is the first Grand Slam title for 18-year-old Hewett, who admitted to being nervous in the closing stages after failing to close out the third set after twice breaking the Frenchmen's serve.
"I got carried away in the moment on a few of the points near the end," he said.
"It was unbelievable. To see so many family and friends here is an amazing feeling. I can't thank the crowd enough for their support."
Gordon Reid and Alfie Hewett become the first Britons to win the Wimbledon men's wheelchair doubles title.