Benzema struck on the half-hour with a badly needed goal, having only scored once in the previous 15 games.
Cordoba did unsettle Real in the second half but Ronaldo hit a trademark 25-yard drive into the corner to settle the game in the final minute.
While they beat Sevilla to win the UEFA Super Cup earlier this month, Real were poor during pre-season and were outplayed by Atletico Madrid in the domestic Super Cup, losing 2-1 on aggregate last week.
"I have always rated the quality of Karim even though he wasn't scoring," coach Carlo Ancelotti told a news conference.
"It would be more worrying if he were in top condition at this stage as is the same for all. We need to speed up the play more. The most important thing was to start with a win which we did even though we had to suffer," he added.
It was a more attacking Real lineup than then one that took the field against Atletico in the Super Cup with new signing James Rodriguez starting ahead of Xabi Alonso in midfield.
After a scrappy start Real began to increase the pressure with Ronaldo and Toni Kroos having shots from the edge of the area which Cordoba keeper Juan Carlos Martin dealt with comfortably.
The pressure eventually told when Benzema opened the scoring, with Mike Havenaar going close at the other end from a corner.
The visitors made it difficult for Real after the restart and Xisco thought he had found an equaliser midway through the second half but was called offside, before Ronaldo ended any thoughts of comeback late on.
Later champions Atletico face Rayo Vallecano. Barcelona beat Elche 3-0 on Sunday.