Bilbao outsprinted breakaway companions Georg Zimmermann (25) of Germany and Australian Ben O'Connor (27), who were second and third respectively.
Defending champion Jonas Vingegaard (26) of Denmark retained the overall leader's jersey as Bilbao moved up to fifth overall from 11th, giving Spain their first stage win on the Tour since Omar Fraile (32) in 2018.
The Basque rider, who will retire at the end of the season, is donating one euro to an association helping replant in local deforested areas for every rider finishing a Tour stage behind him - following an initiative by Mader before he died after crashing in a descent at the Tour de France on June 16.
"It's an incredible feeling. The longer you wait for such a moment the sweeter it is when it happens," said Bilbao.
"I had to win for Gino. I wanted to wait for the biggest moment to win for him. When I crossed the line it was an incredible explosion of feelings.
"It's such a special Tour for me with the race having starting from home (in Bilbao)," the Guernica-born rider said.
"Usually I'm quite cold blooded but today I was very distracted, there was so much at stake."
The stage got off to an explosive start with Tadej Pogacar, closely followed by Vingegaard, on the attack as he kept his rival on his toes while the peloton blew up.
Pogacar's UAE Emirates teammate Adam Yates and his twin brother Simon, fifth and sixth overall respectively, were dropped, as well as fellow top-10 members David Gaudu and Romain Bardet of France.
They hovered two minutes behind the main bunch but as Gaudu's Groupama-FDJ pulled and the peloton's pace eased up, the junction was made before the Col de la Croix Saint Robert (6km at 6.3%).
A seven-man breakaway, featuring Bilbao, eventually took shape and they were joined by a group of seven, among which was former world champion Julian Alaphilippe, 81km into the stage.
In the Cote de Saint Victor la Riviere, Esteban Chaves went solo but was quickly reined in with the peloton controlling the gap.
In the Cote de la Chapelle Marcousse (6.5km at 5.6%), Krists Neilands attacked as the breakaway group was split into three and the Latvian rider had a 20-second advantage 13 kilometres from the finish as temperatures reached 38 degrees Celsius (100.4 Fahrenheit) in Issoire.
Neilands was, however, caught with 3.2km left and Bilbao was the smartest of a group of six to win the final sprint.