Real Madrid Brazilian midfielder Kaka
Argentina winger Angel Di Maria earned both spot kicks, which Kaka coolly converted in the 14th and 54th minutes at the San Mames stadium.
Cristiano Ronaldo came on as a second-half substitute, and capped the win by scoring on a counterattack in the 70th for his league-best 28th goal this season.
The win lifted Madrid to within five points of Barcelona, with the two teams set to play next weekend in the first of four likely "clasico" matches in the coming weeks.
Madrid coach Jose Mourinho experimented with his starting 11 by moving Pepe up into midfield to make way for a rare appearance by Argentine Ezequiel Garay in the center of defense, as Madrid easily shook off last week's home loss to Sporting Gijon by dominating throughout.
Di Maria flashed a shot past the far post 10 minutes before he raced unmarked into the area and was cut down by Bilbao goalkeeper Gorka Iraizoz. Kaka, who has struggled with injuries all season, perfectly placed his shot into the lower left corner of the net.
Gonzalo Higuain had a pair of excellent opportunities to double Madrid's advantage in his first start since returning from injury. But the striker lacked the precision to beat Iraizoz one-on-one in the 26th, and two minutes later missed an empty net with a long-range strike.
Di Maria continued to expose Bilbao's defense, hitting the side netting from close range and barely missing with a lob before the 40-minute mark.
Bilbao had its best chances to level the score late in the first half when midfielder Carlos Gurpegi shot wide and Gaizka Toquero's header drew a save from Madrid goalkeeper Iker Casillas.
The visitors soon made it 2-0 when Di Maria dribbled into the area and was pulled down by Bilbao defender Xavier Castillo. Kaka sent the penalty down the middle after Iraizoz dove early to one side.
Madrid took complete control and Ronaldo, who came on for Higuain in the 62nd, silenced the crowd for good when he got behind the defense, cut back and beat Iraizoz with a right-footer to put him one goal ahead of Barcelona's Lionel Messi in their personal scoring duel.
After Madrid hosts Barcelona on April 16 in La Liga, the pair meet in the Copa del Rey final four days later. And with both holding commanding leads heading into the second legs of their respective Champions League quarterfinal, they are likely to meet twice more in the semis.
Sevilla is at Mallorca later.
On Sunday, third-place Valencia hosts fourth-place Villarreal as the provincial rivals battle for the last Champions League automatic qualifying spot, while Atletico Madrid plays Real Sociedad.
Also, it's: Sporting Gijon vs. Osasuna, Hercules vs.
Espanyol, Racing Santander vs. Levante, and Malaga vs.
Deportivo La Coruna.
Zaragoza hosts Getafe on Monday.