Benzema edges Madrid past Mallorca
Under coach Jose Mourinho, the France international has started most games on the bench, scoring only one league goal before Sunday, and the club have said they are looking for another striker to replace the injured Gonzalo Higuain.
Real struggled to create chances against a gritty Mallorca, who held them to a 0-0 draw on the opening day of the season, and looked in danger of dropping points for a second successive game until Benzema netted in the 61st minute.
The former Olympique Lyon forward picked up a pass on the edge of the penalty area, created space for himself and sent a low shot past Mallorca's Israel goalkeeper Dudu Aouate.
The win prevented Real from slipping further behind leaders Barcelona, who were seven ahead before kick-off after beating Racing Santander 3-0 on Saturday.
Barca have 55 points from 20 matches, Real have 51 in second and Villarreal are third on 42 after a double from Italy striker Giuseppe Rossi gave them a 2-1 comeback win over visiting Real Sociedad in Sunday's late kick-off.
Valencia are two points further back thanks to Saturday's 4-3 home win against nine-man Malaga.
Real Madrid's week had been overshadowed by media reports of a rift between Mourinho and club director general Jorge Valdano and some fans unfurled a banner with a message of support for their Portuguese coach.
With an eye on Wednesday's King's Cup semi-final first leg at holders Sevilla, Mourinho initially rested midfielders Xabi Alonso and Mesut Ozil and Michael Laudrup's well-drilled Mallorca side had the best of the early exchanges.
They came desperately close to a shock lead in the 13th minute when Emilio Nsue sped clear on the right but his powerful effort flew back off the inside of the post with Iker Casillas well beaten.
There were whistles around the stadium when the teams went off goalless at half-time but the introduction of Alonso and Ozil, replacing Fernando Gago and Kaka, gave the hosts more bite and penetration after the break.
After Real had taken a deserved lead, Benzema almost added a second after a goalmouth scramble and La Liga top scorer Cristiano Ronaldo sent a looping header against the bar with just under 10 minutes left.
Real survived a scare at the death when Casillas saved Pierre Webo's close-range strike.
"I decided to make some changes because we are playing another important match (in the Cup) on Wednesday," Mourinho told a news conference.
"Getting to half-time at 0-0 was a very good result for us because we knew we were going to make changes," the Portuguese added. "The changes had their effect and we got a very hard and very important victory."
In Sunday's earlier games, Atletico Madrid's hopes of a Champions League place next season were dealt a blow when they lost 1-0 at struggling Sporting Gijon.
A defensive error allowed striker David Barral to score six minutes after half-time and the defeat leaves Atletico seven points behind fifth-placed Espanyol and follows their elimination from the King's Cup at the hands of city rivals Real Madrid on Thursday.
Surprise packages Espanyol continued their fine run with a 3-1 comeback win at Getafe.
Almeria, who play Barca in the semi-finals of the Cup on Wednesday, climbed off the bottom with a 3-2 win at home to Osasuna and are 18th.