Shelton's service-and-volley game powered him to the opening set but Auger-Aliassime forced a decider with a tiebreak victory in the second and then broke the American's serve to convert a third match point in the final set.
Both players had lost their opening round-robin matches, Shelton beaten by Alexander Zverev and Auger-Aliassime by defending champion Jannik Sinner. The Canadian has given himself a real chance of making the semi-finals with a hard-fought win.
Auger-Aliassime needed medical treatment during his loss to Sinner, but the Canadian brushed off those worries, despite Shelton storming to a 4-1 lead in the opening set.
The American lost his cool when failing to serve out for the set, launching his racket in frustration when Auger-Aliassime made it 5-4, but Shelton broke the Canadian again.
"He was playing much better than me at the start," Auger-Aliassime said.
"It's not often that I get broken twice in the first set indoors."
Shelton forced the only break point of the second set, but in the tiebreak Auger-Aliassime broke immediately to take a 3-0 lead.
Shelton's tiebreak troubles
The American hurt his knee when attempting a quick change of direction, limping through the rest of the tiebreak, but managed to save three set points before a double fault ended his valiant effort. Shelton did not need any medical attention.
The Canadian held break points at 2-1 up in the final set but had to wait until the final game where Shelton was guilty of gifting match points, and Auger-Aliassime did not refuse.
"It was a weird start, but as the match went on I was finding ways to put returns in the court," the Canadian added.
"Once we engaged in the rallies, I felt like I could win more."
Sinner will face two-time winner Zverev in the other Bjorn Borg Group match later on Wednesday, where the Italian can secure a semi-final spot with a win, while Zverev needs a straight sets victory to assure his last-four place.