He eventually emerged after nearly two and a half hours with a 5-7, 6-2, 7-6 (4) win - the first by an American in singles at the event since 2007 - having several times come back from what looked like hopeless positions.
For Cilic, it was the second successive match in which he had led by a break in the final set only to lose, having already done so against Alex Zverev on Sunday.
Sock lost his debut match at the O2 Arena to Roger Federer, but he is now 1-1 in his round-robin group.
"I wouldn't be out here playing, competing if I didn't think I could give myself a chance to win tournaments, to be here", Sock said.
Sock earlier on Monday kept his hopes alive after coming from 4-2 down in the final-set tiebreak to overhaul Wimbledon finalist Marin Cilic 5-7 6-2 7-6 (7-4) and claim his first-ever win at the elite eight-man event.
"I love playing in front of the fans here - they make it feel like home".
Federer will be meeting Zverev later on Tuesday in the other match in the Boris Becker group.
"I think Jack is playing really good", Cilic said. We had a 4am fire alarm in our hotel also.
But Zverev is setting tongues wagging after an impressive year in which he has won five tournaments, and Federer was forced to repel three break points against the German as early as the opening game. "I'm excited to excited to keep winning and keep my hopes alive this week".
Ultimately, the more experienced Croat Cilic, who has a poor record in the end-of-season finals, proved just too erratic, mixing his 35 outright winners with 37 unforced errors.
Hot favourite Roger Federer breezed through a third set decider after being given a fright by world number three Alexander Zverev at the Association of Tennis Professionals (ATP) Tour Finals. It's an awesome atmosphere, you make me feel like home.
"That was a tough one for sure", added Sock who will look to go to 2-1 when he faces Zverev on Thursday (November 16).
Cilic is making his third appearance at the season finale, and was looking to bolster his chances of making the semi-finals for the first time.
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