Without the Manchester United bound Angel di Maria, Real missed the Argentine's energy in midfield but went ahead 15 minutes before half-time when Benzema rose highest to head home Toni Kroos' corner.
Cordoba had their chances to equalise after the break as Iker Casillas denied Ryder Matos and Fidel, but Ronaldo sealed the game in stoppage time with a characteristic thumping finish into the bottom corner from 30 yards.
Real boss Carlo Ancelotti explained his side's below par performance was down to fatigue.
"There was a bit of tiredness because since the 12th we have played every three days and it is not an easy time of the season to do so," Ancelotti said.
"We suffered a little bit, but we controlled the game. We tried to mix possession with playing a bit more direct and I think we were well-balanced."
Cordoba were playing in their first top flight game for 43 years and nearly had a dream start against Real's galacticos when Mike Havenaar pulled his shot just wide from the edge of the area.
Madrid did go in front on the half hour mark as Benzema ended his four-month goal drought at club level with a thumping header to power Kroos' corner in off the post.
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