In a see-saw deciding set, the Croatian fifth seed started quickly and raced to a 3-0 lead before Sock, the eighth seed, steadied himself, cancelling out the break.
Sock will hope to defeat Zverev in the final round-robin meeting in the Boris Becker group in a match which could decide who progresses to the semi-finals while Cilic will face Federer hoping to finish his ATP Finals campaign with his first victory of the week.
Serve dominated the rest of the set but Cilic edged 4-2 up in the tie-break when the ball struck the net chord and Sock failed to dig it over.
"That was a tough one for sure", Sock told Sky Sports' Annabel Croft during his on-court interview.
"It was an interesting day, we had a 4am fire alarm in our hotel also". But I love playing here in London. But I love playing here in London, it's an unbelievable atmosphere, you make me feel like home.
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He may still have to beat Zverev in his final group match to qualify, with the talented German going up against favourite Roger Federer later on Tuesday.
"I'm just excited to win and keep myself alive".
Sock, the American number one, came from nowhere to reach the end-of-season event, reaching London by winning the Paris Masters - and breaking a run of European success at Masters events stretching back to 2010.
The Spaniard quit the tournament on Monday after losing his opening match in the Pete Sampras group to David Goffin.
The ATP Finals feature the top eight fit male singles players and doubles teams who have accrued the most points throughout the 2017 season.
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