He hit a powerful forehand victor to let his intentions known early on and set up three break points in the opening game.
"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. The opening match in the Pete Sampras Group produced a feast of stylish backhands from both players but it was Dimitrov, the first Bulgarian to qualify for the tournament in its 48-year-old history, who overcome some late jitters to prevail.
"I think Jack is playing really good", Cilic said.
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Federer didn't panic, though, and served his way out of trouble, with Zverev missing an early opportunity to snatch the initiative.
The 20-year-old third seed raced into a 4-0 advantage only to see the Swiss charge back to lead 5-4.
There was a jovial atmosphere between the two as they embraced and the youngster suggested a game of "first-to-11, no serves" before Federer got on with his 30-minute hit with towering American Sam Querrey, who is the same height as both Zverev and Marin Cilic.
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The tension returned in the final game of the set, with Zverev facing multiple break points amid distractions from the crowd and celebrating a point he'd thought he'd won only to be overruled by Hawk Eye, but he got the hold nonetheless to force a breaker in the first.
Sock twice served well under pressure to stay in the match and despite falling a mini-break behind in the tiebreak recovered to win five unanswered points to claim victory.
Jack Sock kept his hopes alive in the ATP finals with a courageous display, emerging from the brink of defeat to beat Marin Cilic 5-7 6-2 7-6 (4) in a dramatic if erratic battle of the big hitters at London's O2 Arena on Tuesday. It's been an interesting morning so far, the fire alarm went off at 4am and we had to exit the building. It's an wonderful atmosphere here and I love playing in London and in front of the fans here. "I play a different style and there's always a lot going on, I don't even know what I'm going to do sometimes, it keeps it fun".
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The Swiss star composed himself again to break down a man 16 years younger with six straight games in the third set to make it through to the semi-finals after two hours and 11 minutes of action.