Aritz Aduriz cancelled out a first-half Karim Benzema effort to leave the two teams level in the 65th minute, but Casemiro's close-range effort secured a vital victory for Los Blancos, who have moved five points clear of Barcelona on the same number of matches.
As for Bilbao, Mikel San Jose missed out through suspension and Sabin Merino through injury, but in-demand Inaki Williams was available for the home side. Meanwhile, Aduriz, Ander Iturraspe and Oscar de Marcos also returned to the Basque side's XI following last weekend's 2-0 win at Real Sociedad.
It was the home side that had the first real chance of the match in the eighth minute when former Atletico Madrid attacker Raul Garcia sent a clever volley towards the bottom corner, but Real Madrid stopper Keylor Navas was on hand to make a smart save.
Los Blancos had the ball in the back of the Bilbao net in the 10th minute when Cristiano Ronaldo tapped a Dani Carvajal cross into the bottom corner, but the offside flag was raised, much to the frustration of the Portuguese.
Chances arrived at both ends of the field as the match began to open up, with centre-back Yeray Alvarez firing just wide of the Real Madrid post from inside the box, before Ronaldo was denied by a super save from Kepa Arrizabalaga down the other end.
Ramos was next to come close for Real Madrid as he attempted to continue his excellent scoring run, but the Los Blancos skipper could only head a deep Toni Kroos free kick wide of the post as the score remained goalless.
The pressure was building on the Bilbao goal, however, and Real Madrid finally made the breakthrough in the 25th minute of action when Ronaldo raced down the left and delivered a brilliant cross into Benzema, who found the bottom corner with a super finish.
Casemiro and Raul Garcia had to be separated in an altercation that involved almost all of the outfield players at San Mames, and the referee was busy in the latter stages of the first period due to the ill-tempered nature of the contest.
Benzema and Bale both came close to doubling Real Madrid's lead before the interval, but the former could not convert a deep cross from Marcelo, before the latter saw his effort saved by Kepa.
Bilbao came close to levelling early in the second period when Raul Garcia met a deep cross from Williams, but Navas and Nacho between them just about did enough to clear the danger as the away side's defence was tested once again.
Williams had a super chance to level the scores in the 63rd minute when the in-demand forward met a low cross from the right, but Navas was on hand to make another smart save as Real Madrid kept their one-goal lead intact.
Bilbao were level in the 65th minute, however, when Aduriz nodded home from close range after Raul Garcia had headed a deep Williams cross into the path of his fellow Spaniard.
The response from Real Madrid was impressive though, and the league leaders regained the lead just three minutes later when Casemiro swept the ball home from close range after Bilbao had failed to deal with a corner from Kroos.
Marcelo passed up a super chance to make it 3-1 Real Madrid 15 minutes from time when he fired over the crossbar from close range, before Navas had to get down to keep out a strike from Williams down the other end of the field.
Ronaldo was sacrificed for the final 10 minutes at San Mames as Zinedine Zidane looked to close out the win, and that proved to be the case as the Madrid giants survived a nervy final few minutes to open up a five-point lead at the top of the table.
Real Madrid will return to action after the international break with a home game against Alaves on April 2. Next up for Bilbao, meanwhile, is a trip to basement side Osasuna on April 1.