Murray replaced Novak Djokovic at No. 1 a week ago, and they continue to battle for the year-end top ranking. Murray and Djokovic are competing in different groups in the round-robin portion of this eight-man event. Murray is in the four-man John McEnroe group, while Djokovic is in the Ivan Lendl Group.
The winner and runner-up of each group advance to the semifinals.
Spurred on by the British crowd, Murray had little trouble getting past Cilic to win his 20th straight tournament match. Murray broke serve in Cilic's first two service games to take a 3-1 lead. Murray struggled to hold serve thereafter but held on to win the set and take control of the match.
Murray's next opponent is Kei Nishikori, who beat Stan Wawrinka 6-2, 6-3 in just 67 minutes.
"I felt very comfortable out there," said Nishikori. "I started very well from the first game. I saw that he wasn't playing his best and he was making a lot of unforced errors. I saw a lot of opportunities today, so I tried to be aggressive. Even in the second set, I stayed really tough and played a good match today. It gives me a little confidence."
Wawrinka, the oldest player in the field at 31 years of age, did not have any break-point opportunities on Nishikori's serve.
"It was not a great match, compared to what I can do," said Wawrinka. "It wasn't a great day for myself. In general, I think he played well and put a lot of pressure from the beginning. I don't think I found anything on the court today. I was a little bit slow on everything. I was hesitating a lot with my game, my movement."
The 26-year-old Nishikori lost just five points on his serve in the first set and broke Wawrinka in the fifth and seventh games to take command of the match.
Nishikori finished with 19 winners and Wawrinka had 31 unforced errors.