Tsitsipas reached his first ATP World Tour final earlier this year at a clay-court event Barcelona, losing to Spanish great Rafael Nadal in the championship match.
He'll next face second-seeded defending champion Alexander Zverev of Germany, a 6-3, 6-2 victor over Russia's Daniil Medvedev.
He earned his first match point with a cross-court forehand victor and clinched his two-hour, 18-minute victory with another one, dropping his racquet to the court with joy. "I feel like I'm playing well", Zverev said.
Djokovic was made to work to hold in his first service game and a laboured start soon handed Tsitsipas the break, the Serbian going long and then failing to recover and rescue the opener.
Cilic beat 11th-seeded Diego Schwartzman of Argentina 6-3, 6-2.
"I thought Tsitsipas played very well and he deserved to win without a doubt", Djokovic said.
The Greek teenager, who turns 20 on Sunday, followed up his defeat of seventh seed Dominic Thiem, beating four-time champion Djokovic, seeded ninth, in a first-time meeting.
The second set was a battle as the Swiss, a three-time Grand Slam champion, fought back from an early break down, took a 2-1 lead, with the pair again trading breaks in the fifth and tenth games.
Also, Karen Khachanov of Russian Federation beat 12th-seeded Pablo Carreno Busta of Spain 6-4, 7-6 (3); Ilya Ivashka of Belarus topped American Ryan Harrison 7-6 (5), 6-4; and American Frances Tiafoe edged Canada's Milos Raonic 7-6 (4), 4-6, 6-1. I tried few a things that I didn't try before and I played a bit better.
"The Canadian teenagers are great players", the former world No. 1 said. "I didn't play well". We had a good quality of tennis.
Dimitrov, who last reached a quarter-final in April, added: "Clearly I'm not playing my best tennis, but I'm finding a way and managing to go through those matches".
In the doubles quarterfinals, the top-seeded duo of Austria's Oliver Marach and Croatia's Mate Pavic edged Jean-Julien Rojer of the Netherlands and Romania's Horia Tecau 6-7 (5), 7-5, 10-6.
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