Barcelona can regain the lead when it hosts Mallorca later.
The win at Santiago Bernabéu Stadium extended Madrid's unbeaten run in all competitions to nine matches. It was its fourth straight league win, and the 14th in the last 15 games against Espanyol across all competitions.
Espanyol, winless in six league games, stayed second-to-last in the standings.
"We clearly suffered," Madrid coach Zinedine Zidane said. "We know Espanyol is not a team that deserves to be where it is in the table. But in the end we were patient and earned a deserved victory."
Varane opened the scoring with a low left-foot shot from inside the area in the 37th minute after a well-placed pass by Benzema, who sealed the victory with a shot from close range in the 79th after being set up by Federico Valverde.
"We had to be patient to find the goals," Varane said. "We were struggling with that last touch. Sometimes the strikers are the ones helping us, and today it was my turn to contribute."
Benzema, who has eight goals in his last six matches, is the league's leading scorer with 11 goals, two more than Barcelona's Lionel Messi.
Madrid played the final minutes with 10 men as left-back Ferland Mendy was sent off with a second yellow card for a hard foul in the 83rd.
Madrid entered the match with a slew of injuries, with Gareth Bale, Eden Hazard, James Rodríguez, Marcelo and Lucas Vázquez among those unfit to play.
Zidane started with the Brazilian teenage duo of Rodrygo and Vinícius Júnior flanking Benzema up front. Vinícius Júnior hadn't started for Madrid since the team's 1-0 loss at Mallorca in October.
Zidane had to resort to a three-man back line after Mendy's red card because he had no left back on the bench.
Madrid dominated throughout but Espanyol threatened at times, with Madrid goalkeeper Thibaut Courtois forced to make a good save off a header by Esteban Granero in the first half and Espanyol forward Wu Lei coming close to equalizing in a breakaway early in the second half.
Espanyol loudly called for a second yellow card to Vinícius Júnior for a hard foul late in the first half.
"It's a clear four and in 99% of the time that's a yellow card," Espanyol coach Pablo Machín said. "It could have changed the match."
It was the fifth lost in six league matches for Espanyol, which has nine points from 16 matches, five points from safety.
Espanyol hasn't beaten Madrid at the Bernabéu in 24 consecutive league matches, since 1996. It hasn't kept a clean sheet in the league at Madrid's stadium since a 0-0 draw more than 40 years ago.
Espanyol has the league's worst home record, having won only one point from eight matches. It now has lost three straight away matches.