The world number one was four shots off the lead heading into the last day, but fired a course record-equalling nine-under-par 62 to clinch the title by a single stroke.
Scheffler followed in the footsteps of compatriot Xander Schauffele, who fell away after starting Sunday tied for the lead, by winning the gold medal.
"I'm proud to be American," said the 28-year-old, who was holding back tears on the podium.
"I got emotional the other night watching the gold medal ceremony for the women's gymnastics. I take tremendous pride in coming over here and representing my country.
"It was just very emotional being up there on stage there as the flag is being raised and sitting there singing the national anthem."
Scheffler's win in Paris was his seventh title of a remarkable season in which he has already sealed his second Masters green jacket and second Players Championship trophy.
His dominance was briefly threatened when he was arrested at the PGA Championship in May before charges were dropped, but he quickly bounced back with victories at the prestigious Memorial Tournament and the Travelers Championship the following month.
Scheffler is widely regarded as a cool customer, but says emotions often get the better of him after a win.
"Typically I'm fairly emotional as it is. I think with this tournament, there wasn't much time in between winning, and then you kind of just get ushered through everything," he said of having little time to himself before the medal presentation.
"I remember after the Masters this year, I took a bathroom break but really I just wanted to be by myself for a minute. I cried pretty good after that one."
Gold was 'slipping away'
Scheffler started quickly with three successive birdies, including a chip-in on the second hole, to close the gap on overnight leaders Schauffele and Jon Rahm.
The birdies dried up and a missed six-foot putt on the par-five ninth was his sixth straight par.
But Rahm, who was four shots clear after 10 holes, started to falter and Scheffler was ruthless, romping home with a six-under back nine of 29.
"I felt like at the time it was definitely slipping away," Scheffler said of his miss on the ninth.
"It can feel pretty frustrating at times to feel like you're doing the right things and not getting anything out of it."
Britain's Tommy Fleetwood was tied for the lead with Scheffler before a bogey on the 17th.
He had to hole out from off the green for birdie on the final hole to force a play-off for gold.
After his chip did not drop, Fleetwood holed a nervy six-foot putt to clinch the silver medal ahead of Hideki Matsuyama who took home bronze for Japan.
"I was obviously trying to catch Scottie and finish in front of him," said Fleetwood.
"Once that chip had missed, I had to sort of turn my attention to try and make sure I hole the next putt as well.
"I've certainly loved that back nine. Well, the whole of today but the back nine, in contention, playing very, very well. The leaderboard changed quite a bit, and it was amazing to be a part of."
