The top-seeded Canadian lost in just an hour and 20 minutes as Rinderknech sealed a 6-4 6-4 victory to seal his place in the semi-finals in Doha.
Shapovalov struggled throughout, making nine double faults and having to face 10 break points, of which he managed to save eight.
Rinderknech will now face Nikoloz Basilashvili after the Georgian fought back to beat Marton Fucsovics 1-6 6-3 7-5.
The other semi-final will see Karen Khachanov take on Roberto Bautista Agut, with the Russian also coming from a set down in beating Marin Cilic 3-6 6-3 6-4.
Bautista Agut followed up his 6-0 6-1 victory against Andy Murray by again winning comfortably against fellow Spaniard Alejandro Davidovich Fokina, scoring a 6-1 6-1 success.
The number two seed won 74 per cent of points when landing his first serve as he sealed victory in an hour and 17 minutes.
At the Open 13 in Marseille, Rotterdam champion Felix Auger-Aliassime increased his winning streak to six matches after beating French veteran Jo-Wilfried Tsonga 7-6 (7-3) 6-2 to head through to the quarter-finals.
"It was a special moment," Auger-Aliassime said after his win. "When the match was coming to the end, I knew it was going to be a special moment, a full-circle moment. He was my top idol growing up and I saw myself in him at times."
Auger-Aliassime will play world number 49 Ilya Ivashka next, while top seed Stefanos Tsitsipas also went through after a 6-3 7-6 (7-4) win against Hugo Gaston. The Greek star will play Russian Roman Safiullin in the last eight.
Ninth seed Benjamin Bonzi beat fellow Frenchman Pierre-Hugues Herbert 6-4 6-3 and will go up against Aslan Karatsev in the next round, while Lucas Pouille overcame Radu Albot 6-3 3-6 7-6 (7-2) and will tackle Andrey Rublev next.