- Karim Benzema (72' Pen)
- Oihan Sancet (49')
Earlier on Sunday, Madrid announced that Ballon d'Or winner Benzema will leave the club as a free agent in the close season after 14 trophy-laden years, with the French striker linked with a move to Saudi Arabian side Al Ittihad.
"One of the best players in the world, one of the best on the planet. A complete footballer, a very nice, humble, serious person," head coach Carlo Ancelotti said in a postmatch news conference.
"We can't be happy, but we have to respect his decision, he's earned the right. We all have to thank him. What he's done for this club is legendary, unforgettable, it will be in this club's history forever."
After Thibaut Courtois saved a Mikel Vesga penalty early in the first half, Bilbao took the lead in the 49th minute through Oihan Sancet, who stabbed a tame effort at the goalkeeper before rifling in the rebound for his 10th league goal of the season.
Real levelled in the 72nd minute after Eder Militao took an elbow to the face from Yuri Berchiche in Bilbao's penalty area, with Benzema stepping up to calmly slot home the spot-kick before receiving a standing ovation from the fans.
Ancelotti said Benzema's decision to leave was "surprise" but added the club should not be rushed into finding a replacement.
"It surprised us, it was a surprise for everyone but we have to understand it, he's thought about it. It's part of the transition at this club which is ongoing, we have time to think about what we'll do, the squad will be competitive next season.
"The club will prepare an event over the next few days," he added. "Karim is very shy, his reaction [Sunday morning's statement] was one of a character like him, he doesn't like the spotlight. I think it was the farewell of a great player and a legend."
Real held onto second place despite the draw, as 10-man Atletico Madrid were held 2-2 at Villarreal following Jorge Pascual's 92nd-minute equaliser. Diego Simeone's side finished the season with 77 points, one behind local rivals Real.
- Isidro Díaz De Mera Escuderos