Mario Mandzukic missed two clear chances in the first half for Atletico, whose Spanish Super Cup victory last Friday over Real Madrid appeared to take its toll as the players looked fatigued in the second half.
"The team played very well, they could have scored in the first half and later what happened is normal. We played Tuesday and Friday last week and put in all that effort," Atletico Madrid coach Diego Simeone said in a news conference.
"We were better in the first half and had three clear chances and Rayo improved in the second half and caused us problems," added Simeone.
Simeone was absent from the touchline due to a ban from last season and he has been given a further eight-match sanction on Monday for his dismissal and angry reaction in the Super Cup.
Atletico were missing in-form Raul Garcia in attack through suspension and was repalced by Raul Jimenez while midfielder Arda Turan was still injured.
Mandzukic first failed to hit the target as he stretched for a cross with the goal gaping and then he shot tamely at the keeper Cristian Alvarez, but for most of the time Rayo held their own.
In the second half Rayo began to dominate with Javier Aquino in particular a threat but if Atletico were lacking their usual attacking menace they were typically secure at the back and did not give Rayo many scoring opportunities.
It was a more attacking Real lineup with James Rodriguez starting ahead of Xabi Alonso in midfield and they failed to link up well in a scrappy opening spell.
Real began to raise the tempo with Cristiano Ronaldo and Toni Kroos having shots from the edge of the area which Cordoba keeper Juan Carlos Martin dealt with comfortably and the pressure told with Benzema's goal after 30 mintues - his second goal in 15 games.
The visitors made it difficult for Real after the restart and Xisco thought he had equalised midway through the second half but was called offside.
They gave Real few chances but still Ronaldo showed his quality by taking full advantage of the opening he had with a 25-yard drive into the corner to settle the game in the final minute.
"I have always rated the quality of Karim even though he wasn't scoring," coach Carlo Ancelotti told a news conference.
"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," added Ancelotti.