World No. 52 Bedene survived a break point early in the opening set before breaking world No. 6 Tsitsipas' serve to take the lead. The Slovenien held his serve throughout the match and edged the second sets to win 7-5, 6-4.
"To win against a top 10 player feels great," Bedene said in a post-match interview. "It's only my second time.
"I am serving really well lately. I'm trying to stay focused on my serve, especially against big players with big serves (where) you don't get many chances. I guess I served well in the important points and kept my cool."
Bedene, 30, will now face Canada's Felix Auger-Aliassime in the quarterfinals.
Information from Reuters contributed to this report.