The favourite was much improved in the second set but still could not shake off Tsitsipas, needing to save two late break points as his opponent closed on victory. "So just all in all, it wasn't that great of a match".
Djokovic, a four-time Rogers Cup champion and former world No. 1, returned to the top 10 in mid-July after missing several months due to a right elbow injury.
After losing the first set 6-1, wild card Wawrinka took the second 7-6 (7/2) to force a third.
Tsitsipas would not be brushed aside, though, and a huge shot down the line sealed an early break in the third, before fighting back in a sublime hold that prompted Djokovic to smash his racket. It was all square on 4-4 when Tsitsipas attempted to break the world number ten's service, however Djokovic endured and led the set to a tie-break, which he eventually won, to bring the game to a draw (6-7).
Milos Raonic of Thornhill, Ont., was scheduled to follow Djokovic's win on centre court but that match was also delayed.
Second seed Caroline Wozniacki of Denmark also crashed out after she lost 5-7, 6-2, 7-6 to Belarus' Aryna Sabalenka.
Dimitrov beat Frances Tiafoe of the United States 7-6 (7-1) 3-6 7-6 (7-4). "I did some things that he didn't expect me to do".
"I was serving well so that break was everything", Tsitsipas said.
"I'm going to get into Cincinnati, the only [Masters] 1000 event that I didn't win in my career, and hopefully I can have a good tournament", ATP website quoted Djokovic, as saying. There was a combined two doubles titles between them as Anderson and Djokovic each had won once before.
He defeated seventh-seeded Dominic Thiem of Australia earlier this week and will be looking to avenge a semi-final loss to Zverev last week in Washington.
The tennis player further credited Tsitsipas for the victory and praised him for displaying commitment and discipline on the court.
Garcia proved too strong for Sharapova with a 6-3, 6-2 win and will take on top seed Simona Halep of Romania in the quarter-finals of the Women's Tennis Association Premier 5 tournament. Nadal opens his 2018 Rogers Cup campaign in Toronto on Wednesday.