The Frenchman, who appeared marginally offside, nodded home an Isco cross after 55 minutes against a well organised Granada side that attacked with pace.
Real now have 10 points from four games and are a point ahead of Atletico Madrid and champions Barcelona, who have won their opening three matches and face Levante at home on Sunday.
Cristiano Ronaldo had eight goals from his previous two games but he and his Real team mates struggled to beat keeper Andres Fernandez.
Granada will feel they deserved more, but poor finishing especially from Isaac Success cost them dearly.
“It was a strange game. We had chances but they did also and in the end we were able to get the goal that helped us a lot and these are three important points,” Real full back Marcelo told reporters.
Real were missing the injured Gareth Bale and James Rodriguez and it was Lucas Vazquez who was picked on the right wing while defensively they were also without centre half Sergio Ramos.
Real suffered an early scare with Cristiano Biraghi bursting down the left wing and his cross was cleared in front of goal by Marcelo.
It was an indication of the problems that Real were to have at the back with Success having a drive well saved by Keylor Navas.
At the other end Ronaldo and Luka Modric, in particular, had efforts saved by Fernandez and it was a similar story after the break.
Success blasted wide with the best chance when in front of goal after 51 minutes and shortly after Granada were made to pay for their profligacy with Benzema’s header.
Atletico Madrid substitute Angel Correa made the breakthrough in his side’s hard fought 2-0 victory at Eibar.
Diego Simeone’s side had made little impact against a determined defence but the introduction of Correa and Fernando Torres in the second half changed the game.
Torres set up Correa for the opener after 62 minutes and then the Argentine youngster was the supplier for Torres who dinked the ball over keeper Asier Riesgo for the second after 78 minutes.
“He (Correa) is a dangerous player especially in front of goal,” Simeone told a news conference. “The team needed someone who could create an opening against a strong defence and the substitutions worked out well.”
Valencia’s stuttering start to the season continued as they were held 0-0 at home to 10-man Real Betis and have just one league win from four games.
A lethargic display saw them unable to break down Betis even after Daniel Ceballos was sent off at the start of the second half.
An injury time header from Hernan Perez gave Espanyol a 3-2 victory away to ten-man Real Sociedad.
Imanol Agirretxe put Sociedad ahead but keeper Geronimo Rulli conceded a penalty and was sent off before half time. Gerard Moreno scored from the spot-kick and Enzo Roco headed Espanyol ahead after 70 minutes.
With four minutes to go Jonathas appeared to have earned Sociedad a draw with a clinical finish but there was still time for Perez to head in from a corner.
(Reporting by Tim Hanlon; editing by Toby Davis/Clare Lovell/Alan Baldwin)