Real still seem to be finding their feet under new coach Carlo Ancelotti and, although they dominated the first half, they lost control in a poor-quality second period at Granada's Los Carmenes stadium.
France forward Benzema, who also scored in the 2-1 opening day success at home to Real Betis last weekend, skewed the ball into the net to give Real the lead after an unmarked Cristiano Ronaldo lost control of an Angel Di Maria cross.
Roberto palmed an Isco effort on to a post in the 62nd minute and Ronaldo sent a low strike narrowly wide late on but Granada finished strongly and had several chances in the final minutes.
Italian Ancelotti, who told reporters it was crucial in the early part of the season not to drop points, has much to ponder before Athletic Bilbao visit the Bernabeu on Sunday on matchday three.
The victory puts Real on six points, the same as champions Barcelona, Atletico Madrid, Bilbao and Villarreal. Barca, 1-0 winners at Malaga on Sunday, top the standings on goal difference, with Real in fifth.
"This sometimes happens in the early part of the league campaign and we are just happy with the three points," Real's Spain defender Alvaro Arbeloa said in an interview with television broadcaster Canal Plus.
"We played a very good 45 minutes, we had control, we created chances," the former Liverpool player added.
"This is only the second matchday and obviously you have to be patient, we have just begun, with a new coach and new players.
"It takes time to get to your top level but I don't think we suffered all that much tonight."
FAN HERO
The build-up to the match was dominated by speculation about whether Ancelotti would restore Spain goalkeeper and captain Iker Casillas to the starting lineup.
Casillas, a hero to Real fans, fell out with former Real coach Jose Mourinho last season and was replaced by Diego Lopez and Ancelotti has surprised many by sticking with the former Sevilla keeper for Real's opening two La Liga matches.
Supporters at the Bernabeu chanted Casillas's name during a friendly last week and even the Granada fans were shouting "Iker, Iker" at one point during Monday's match.
"It's a matter for the coach and we will back whatever he decides," Arbeloa said.
"I have absolute respect for both keepers and whoever plays I will support and I think my team mates feel the same.
"It would be better if the fans did the same and supported the keeper the coach selects."
(Editing by John Mehaffey)
