Groves secured the victory in a tight finish, beating Danny van Poppel (Bora-Hansgrohe) and Tim Merlier (Alpecin-Deceuninck) to the line.
Groves came up on the right side from behind Daniel McLay (Arkea-Samsic) as he sprinted past him and Juan Sebastian Molano (UAE Team Emirates) inside the final 50 metres. Van Poppel and Merlier came off Groves' wheel to finish second and third respectively.
The Australian's victory came as a relief to the team after their leader Simon Yates was forced to pull out of the race following a positive COVID-19 result ahead of Wednesday's stage.
"It feels fantastic," Groves said. "This morning with the news of Simon going positive for COVID-19 all the boys were disappointed. It's the best way to bounce back after such bad news.
"We set up early for this wind. We thought there'd be crosswind earlier but there was nothing until the final 4km where we were the first team. The guys did a perfect job keeping me in front and fresh.
"Maybe with about 50-100m to go I was fresher than the other guys who had to do efforts from behind to move up. Thankfully I was in the right position and got the gap to step out and do my sprint.
"Being my first Grand Tour win for sure it takes the cake."
Race leader Remco Evenepoel (QuickStep-AlphaVinyl) kept his red jersey after finishing safely in the peloton, while Mads Pedersen of Trek-Segafredo stayed on top of the points classification.
Earlier, Alaphilippe was forced to abandon the stage with 64km remaining when he slid out on a right-hand bend while riding towards the head of the peloton with Evenepoel and the rest of the Quick Step-AlphaVinyl team.
The French rider left the stage in an ambulance with a suspected broken collarbone and the crash casts doubt over the defence of his title at next month's World Championships in Wollongong.
