On a weekend dominated by Saturday's "Clasico" between leaders Barcelona and second-placed Real Madrid, which ended 1-1, the Basque club (48 points) climbed into fifth above Sevilla (46), who lost 1-0 at Getafe on Saturday.
Their other direct rivals for a Europa League berth, holders Atletico Madrid (46) and Espanyol (45), drew 2-2 in Barcelona and are seventh and eighth respectively with six games left.
Bilbao had defender Xabi Castillo sent off for a second yellow card in the 57th minute in Pamplona, shortly after striker Kike Sola had fired the home side in front.
Osasuna were unable to make their numerical advantage count and Bilbao's Spain striker Fernando Llorente levelled with a header with just over 20 minutes left.
After Osasuna defender Sergio Fernandez was sent off for a second yellow in the 88th minute, Iker Muniain pounced on a defensive howler by goalkeeper Ricardo Lopez to curl the ball into the empty net from wide on the left.
"We only created two chances and we were 100 percent efficient," Bilbao coach Joaquin Caparros told a news conference. "The team has desire and soul and they showed that on the pitch."
SURPRISE PACKAGE
In Barcelona, Atletico were ahead in the second minute after an error by Espanyol defender Ernesto Galan allowed Koke to score into an empty net.
Pablo Osvaldo equalised with a deflected shot shortly before halftime before Sergio Aguero intercepted an attempted backpass in the 49th minute and lifted the ball superbly over onrushing Espanyol goalkeeper Carlos Kameni.
Osvaldo levelled again nine minutes later but neither side found the firepower or creativity to engineer a winner.
In the day's earlier games, surprise package Levante, promoted at the end of last season and unbeaten in eight games since mid February, strengthened their grip on ninth thanks to a 2-1 win at home to struggling Hercules.
With only 12 points separating 11th from 20th in a congested bottom half of the table, Deportivo La Coruna secured a crucial 2-0 win at home to Racing Santander, while Real Sociedad beat visiting Sporting Gijon 2-1 in San Sebastian.
Almeria remain rooted to the bottom on 26 points after they were beaten 3-0 at home by third-placed Valencia on Saturday.
Hercules have 30 points in 19th and Real Zaragoza, who play at fourth-placed Villarreal on Monday, are 18th on 33.
