Our tennis columnist, Dan Weston, previews the two matches.
Zverev missed out on a semi-final berth on his debut at the season-ender past year and will need to beat John Isner on Saturday to have a chance of sticking around for the weekend.
On the first day of play on Sunday, Kei Nishikori handed Roger Federer a surprise defeat as he won 7-6 (4), 6-3 while Kevin Anderson made a winning debut at the tournament with a 6-3, 7-6 (10) victory over Dominic Thiem.
Despite the match likely to be dominated by servers, I still only make a first set tiebreak around a 40% chance, making the 5/6 with the Sportsbook about this eventuality a rather short price.
But first, Djokovic aimed to hold up his end of the deal by beating Zverev, for if the youngest man in the tournament should reverse their recent Shanghai semi-final result, and Cilic beat Isner, it would be the young German who sealed his place in the semis.
Djokovic netted a routine forehand when serving at 4-4 30-30, offering Zverev a break point that he did not convert.
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After defeating Cilic in his opening match, Zverev still has a chance to become the first German since 2003 to advance to the semifinals of the season-ending competition.
He could qualify for his eighth semi-final in the tournament if he beat Alexander Zverev and if John Isner then beat Marin Cilic.
It was Zverev's 55th match win of the year, more than any other player on the Tour.
"I don't think it was breathtaking tennis, but a win is a win", said Djokovic. On the second, he slowed things down, took a deep breath, bounced the ball 17 times-and 11 more on the second serve-and held, 5-4.
Djokovic, who helped Serbia win the title in 2010, is not convinced the two new events can survive.
The second set started quickly, with both players dominant on their serves.
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Cilic's lead didn't last long, however, as an unforced error of his own gifted Zverev the break to bring it back to 4-4.
"I thought I played well from midway through the second set, I started to swing through the ball but I haven't served that great". You have to give him credit for that, qualifying for World Tour Finals two years in a row.
The ATP Finals is contested by the eight players who have accumulated the most ranking points over the season and is in a round-robin format, with the best four players reaching the semifinals.
"I am going to enjoy some quality time with my family and then get ready for the last game in the group". And a double-fault from Zverev in the next game gifted Djokovic the set after 48 minutes.
"Unfortunately today hasn't been great in terms of that", said Djokovic, when asked about his health.
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