The victory came just hours after defending champions Atletico Madrid slumped to a fourth consecutive league defeat, going down 2-1 at Granada, extending their worst run in the competition since
A total of 12 players between Madrid and Bilbao missed the game after testing positive for COVID-19 — including the visitors' Luka Modric and the hosts' goalkeeper Unai Simon.
Benzema opened the scoring with a superb curling strike four minutes into the match at San Mames before he struck again in the seventh for his league-leading 15th goal of the season.
The Frenchman also became just the sixth player — to score 30+ La Liga goals in a single calendar year this century.
Oihan Sancet replied for Bilbao to close out a wild first 10 minutes of the game.
Inaki Williams went close to grabbing an equaliser with a half-hour to play, but a defence anchored by Eder Militao held on to increase Madrid's lead to eight points over second-place Sevilla — who have a game in hand over Los Blancos.
The outbreak in Madrid's squad gave Eden Hazard, Eduardo Camavinga and Nacho Fernandez the opportunity to earn rare starts.
"I want to highlight the players who hadn't played much were ready to go, that is the strength of a team that wants to compete," coach Carlo Ancelotti, whose side enters the winter break on a 15-game unbeaten streak in all competitions, said.
Atletico had not lost more than two league games in a row since Simeone became coach in December 2011 but his month have suffered successive defeats to Mallorca, Real Madrid, Sevilla and Granada.
They took the lead through Joao Felix only three minutes in but the hosts hit back to level through Darwin Machis in the 18th and veteran striker Jorge Molina scored from a cross to complete the comeback in the 61st.
Atletico are fifth, 17 points adrift of Madrid.
"We are not getting the wins like we used to," Simeone said.
"Many times we won games we did not deserve to win, now we are losing games we deserved to win."
Molina was coming off a record-setting match on Sunday when he became the oldest player in Europe's top five leagues to score a hat-trick at 39 years and 241 days in a 4-1 win over Mallorca.
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