Struff, who has yet to win an ATP singles title in his decade-long career as a professional, was on top for most of the match against the 21-year old Greek.
He served for the match in the second set but Tsitsipas managed to stay alive by taking the second set on a tiebreak.
However, the German refused to give up and sealed the win to set up a last 16 meeting with ninth-seed Daniil Medvedev, who beat Frenchman Benoit Paire 7-6(2) 6-1.
Among the other surprises on Wednesday, world number five Kei Nishikori was beaten 7-6(2) 6-4 by Japanese qualifier Yoshihito Nishioka.
World number one Novak Djokovic and third seed Roger Federer advanced to the last 16 on Tuesday. (Reporting by Andrew Downie, editing by Peitha Sarkar)
Chris Froome (Stage 3, 2015)
James Toseland (1/2)