Sixth seed Tsitsipas was scheduled to face Kohlschreiber on Wednesday, however, the German veteran succumbed to a muscle strain.
Greek star Tsitsipas - last year's semi-finalist - will face either 32nd seed Milos Raonic or Cristian Garin for a spot in the fourth round at Melbourne Park.
Tsitsipas announced himself after stunning Roger Federer en route to the Australian Open semis in 2019, the 21-year-old's run ended by 19-time grand slam champion Rafael Nadal.
After a delay of more than four hours because of rain in London, Berrettini overcame Evans 7-6 (7-5) 6-3 to reach his third ATP grass-court semi-final.
The Italian hit 13 aces and won 81 per cent of his first-serve points against Evans to set up a last-four clash with Alex de Minaur.
Berrettini improved his win-loss record to 24-6 for the season and laid out his ambition to walk away from the tournament with the trophy.
"I didn't serve that well, but I was returning well and I just played better in the last few points of the tie-break," he said.
"After that, I felt more confident. The conditions were really tough, windy and cold, so I took time to adapt a little bit. I am pretty happy with my performance.
"The court condition was really good. I expected slippery conditions, but it was like yesterday.
"I came here to win the tournament, that is my goal. Now I am two steps away. I am happy with the way I am playing, and my mental attitude is really good."
Up next for Berrettini is Australian De Minaur, who came from behind to defeat Marin Cilic 3-6 6-3 6-4.
The 22-year-old won 73 per cent (22/30) of his second-serve points and saved six of the seven break points he faced as he moved to 16-12 for the season.
In the battle of the British players, Cameron Norrie beat Jack Draper 6-3 6-3, while Denis Shapovalov was leading Frances Tiafoe 6-3 when their match was suspended due to fading light. They will resume on Saturday.
At the Halle Open, Andrey Rublev reached his sixth ATP Tour semi-final of the year thanks to a 7-6 (7-4) 6-2 win over 2011 champion Philipp Kohlschreiber.
"I am happy with my performance to reach the semi-finals for the first time," Rublev said. "The first set was really tough. He was 3-0 up in the tie-break and I came back, which was the key.
"After the first set, I think he mentally went down and I was pumped up. I hit a couple of good returns in the first game of the second set."
Russian Rublev will face Nikoloz Basilashvili in the last four after the Georgian defeated Lloyd Harris 6-4 7-6 (7-5).
In the day's other quarter-finals, Felix Auger-Aliassime beat Marcos Giron 6-3 6-2 and Ugo Humbert overcame Sebastian Korda 6-2 6-7 (5-7) 6-4.
Shapovalov was beaten by fellow Canadian Vasek Pospisil in his opening match at the Oden Sud de France last week after falling at the first hurdle at the Australian Open.
The world number 16 suffered another setback on Monday, going down 6-3 7-6 (7-3) to the unseeded Dimitrov.
Dimitrov was beaten by outsider Gregoire Barrere in his first match in Montpellier last week but claimed a fourth tour-level victory of the year after saving all three break points he faced against the eighth seed.
The former world number three could face another Canadian in the form of Felix Auger-Aliassime or German Jan-Lennard Struff in the second round.
Roberto Bautista Agut, the sixth seed, came from a set down to defeat Marton Fucsovics, while fellow Spaniard Pablo Carreno Busta downed Adrian Mannarino 7-5 6-7 (3-7) 6-4.
Dan Evans advanced in the final match of the opening day of the tournament, seeing off qualifier Philipp Kohlschreiber 6-3 7-5.
World number 84 Ruusuvuori saved a set point in the first-set tie-break on his way to a 7-6 (8-6) 6-4 victory and will now face Ilya Ivashka for a place in the semi-finals.
There was nearly another upset as home hopeful Alexander Bublik, seeded second, had to recover from a set down to overcome Miomir Kecmanovic 2-6 6-3 7-5.
Carlos Taberner awaits Bublik in the next round after the Spaniard beat Egor Gerasimov 5-7 7-6(5) 7-5.
Kwon Soon-woo and James Duckworth saw off Dusan Lajovic and Filip Krajinovic respectively in straight sets to remain a course for glory.
At the Moselle Open in Metz, meanwhile, Gael Monfils took out Philipp Kohlschreiber 7-6(2) 6-4 to reach the quarter-finals on home soil.
Tournament favourite Hubert Hurkacz made light work of 2016 champion Lucas Pouille with a 6-2 6-3 victory to set up a meeting with Andy Murray in the quarter-finals.
Elsewhere on Thursday, German qualifier Peter Gojowczyk proved too strong for Karen Khachanov and will now face Marcos Giron, who upset fourth seed Alex de Minaur.
