Italian 21-year-old Sinner, ranked 14 in the world and fresh from his Montpellier title last weekend, came through 6-4, 6-3.
It was only his second win over world number three Tsitsipas in seven meetings and avenged a five-set loss to the Greek player in the last-16 at the Australian Open last month.
Sinner will face Swiss veteran and three-time major winner Stan Wawrinka for a place in the semi-finals.
Highly-regarded 19-year-old Rune was trailing American-born Dutchman Brouwer 6-4, 4-0 when he quit their second round tie.
Rune retired after a medical timeout failed to fix his physical problems.
Brouwer will face compatriot Tallon Griekspoor in Friday's quarter-finals.
Defending champion Felix Auger-Aliassime of Canada eased past Gregoire Barrere to reach the last eight.
The 22-year-old from Montreal dispatched the last French player standing 6-4, 6-3 to set up a meeting with sixth seed Daniil Medvedev, who beat Botic van de Zandschulp 6-2, 6-2, saving all six break points he faced.
Number three seed Auger-Aliassime secured his place in the last eight on the back of a solid first service, firing nine aces.
Medvedev will be favourite to win their encounter having won all of his four meetings with the Canadian, with all but one going the distance.
"Felix is a top player. He won here last year and played amazing at the end of last year. Looking forward to a tough match and hopefully I can be at my best," said Medvedev.
Australia's Alex de Minaur, who upset second seed Andrey Rublev in straight sets on Wednesday, plays Grigor Dimitrov in Friday's other quarter-final.
