Newly crowned NBA defensive player of the year Wembanyama muscled his way to 12 blocks - a record in a postseason game - but it was not enough to thwart a resilient performance from the sixth-seeded Timberwolves.
Minnesota ace Edwards, who missed the final two games of the Wolves' first-round series win over Denver with a left knee injury, scored 18 points from the bench despite clearly being short of full fitness.
Edwards' effort was backed by 21 points from Julius Randle while four other Minnesota players cracked double figures in a precious opening victory on the road in the best-of-seven Western Conference semi-final series.
"We just want to win ball games, that's all we want to do as a group, no matter who it is, we don't care," Edwards said.
"I may not be as athletic as I usually am, but I got to be able to box out and make those small plays and win a big-time game," he added.
San Antonio were left reflecting on an agonising finish which saw Julian Champagnie's attempted buzzer-beater to snatch victory bounce off the rim as the clock expired.
Wembanyama's monster defensive performance also included 15 rebounds with 11 points in what turned out to be a losing effort.
Dylan Harper led the Spurs scoring with 18 points off the bench on a night when seven San Antonio players finished with double-digit points tallies.
Wembanyama vowed an improvement when San Antonio attempt to level game two of the series on Wednesday.
"Obviously we have to be better," he said.
"We need to figure out in the next 48 hours what we can do better. And I've got no doubt that we will. I would trust us," the Frenchman said.
"It's a lot on me because obviously my game wasn't feeling good tonight and that's where the team is going to look for me," Wembanyama added. "If I had been better, if the offensive leaders of our team had been better, it would have been different."