The Italian, who needs to win the season-ending event for the top eight players and hope Carlos Alcaraz doesn't reach the final to also claim the year-end No.1 ranking, enjoyed a 7-5 6-1 win over the top Canadian.
No.8-ranked Auger-Aliassime appeared slowed by a left calf issue that arose near the end of the first set and was twice treated by a trainer during the second.
"I hope it's nothing too serious," Sinner said.
"I wish him obviously a very speedy recovery and hopefully he is back to 100 per cent physically."
Earlier on Monday, last year's losing finalist Taylor Fritz beat Lorenzo Musetti 6-3 6-4 to win his opening round-robin clash in Turin.
The match was the second in Alex de Minaur's Jimmy Connors Group following the Australian's loss to top seed Alcaraz on Sunday.
Sinner won the title last year without dropping a set and hasn't lost at the event since the 2023 final to Novak Djokovic.
The 24-year-old has won all four of his meetings with Auger-Aliassime this year, including a US Open semi and the recent Paris Masters final.
"Obviously winning the first match is very important in this competition and this format ... so I'm happy to overcome a very tough test today," Sinner added.
Both he and Alexander Zverev lead the Bjorn Borg group with one win each after Zverev beat Ben Shelton on Sunday.
Musetti, a late replacement after the withdrawal of seven-time winner Djokovic, began brightly but started to tire despite the backing of a largely partisan crowd at the Inalpi Arena.
Beaten by Djokovic in an almost three-hour final in Athens on Saturday, Musetti managed to claw back four break points in the second set opener, but two double faults in his next service game gave Fritz the decisive break.
Fritz served nine aces in the second stanza, serving to love on three occasions, en route to an impressive win.
"The whole second set I played well and had a lot of chances to break that I didn't get," Fritz added.
"I'm really happy I was able to serve it out there and he didn't come back to ruin it."
Musetti will now play de Minaur on Tuesday with both players needing a win to maintain a chance of qualifying for the semi-finals, while Fritz faces Alcaraz.
With AP.
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