The victory has seen Madrid return to within four points of second-placed Atletico Madrid in the table, while Bilbao remain seventh, three points off sixth-placed Valencia, who now have a game in hand.
Marcelo's energy down the left was a key feature of Madrid's threat inside the first 10 minutes, but Bilbao were also having their moments on the break with Inaki Williams threatening to expose Jesus Vallejo.
Marcelo had the game's first shot in the 14th minute, but his effort from inside the Bilbao box was too high.
Inside a Bernabeu which was far from at capacity, chances were few and far between in the first 25 minutes. It was a decent game in terms of tempo and intensity, but both sides were struggling to carve out opportunities.
Having won four of their five in the build-up to this one, Bilbao arrived full of confidence and the fact that they were prepared to commit at least four players to each attack was an indication of their belief.
Truth be told, it was a poor first 45 minutes, with neither team doing enough to deserve to be ahead and the second half started with the same group of 22 players.
The breakthrough arrived less than two minutes into the second period, though, when Benzema headed an Asensio cross into the back of the net for his 28th goal of the campaign.
Toni Kroos - back from illness - almost scored a second moments later, but the German midfielder saw his close-range attempt deflected just wide of the away side's post. Inigo Martinez then struck over the Madrid crossbar from distance during a much better period of the match.
Benzema had another chance just before the hour, but the striker's effort just missed the far post during a spell where Vazquez and Asensio were seeing much more of the ball in forward positions.
The home side doubled their lead moments after the substitutions, though, with Benzema rising highest inside the Bilbao box to head a Luka Modric corner into the back of the net.
Two more chances fell to Williams and Bale at either end, but Navas kept out the former's close-range header before the latter chipped over the crossbar from a good position.
Madrid were not finished, though, and scored a third in the 91st minute when Benzema completed his hat-trick by lifting the ball over Herrerin after the Bilbao goalkeeper had vacated his goal.
Next up for Zinedine Zidane's side is a trip to Getafe on Thursday night, while Bilbao are back in action away to Leganes 24 hours earlier.
REAL MADRID (4-3-3): Navas; Carvajal, Varane, Vallejo, Marcelo; Kroos (Isco 70'), Casemiro, Modric; Lucas (Bale 70'), Benzema, Asensio (Brahim 84')
ATHLETIC BILBAO (4-2-3-1): Herrerin; Lekue, Alvarez, Martinez, Berchiche; San Jose, Benat; Muniain, Garcia (Aduriz 76'), Cordoba (U Lopez 76'); Williams