Spain had been trailing at half-time, but Villa equalised with a shot from the edge of the area before converting a penalty to maintain Spain's 100% record in Group I.
His efforts on the night took his international goal tally to 46, surpassing Raul's record, but Villa preferred to put the focus on the result.
"I am happy because the goals helped the team get the win," he said in AS. "The important thing is that we are close to qualifying for the tournament.
"I really wanted it, mostly because it was a very difficult game, which we did not start as we expected. We had chances, but luck and their keeper were against us."
Villa ran towards Pepe Reina on the bench after scoring the goal and revealed he had dedicated the landmark to the Liverpool goalkeeper.
He said: "The goals are dedicated to all the team-mates, all the coaches, all partners and friends I had during my career - but long ago I promised Jose, who is a friend of mine, that the goal which overtook Raul would be for him."
Spain coach Vicente Del Bosque was happy to give praise to Villa after he etched his name into the record books.
"It is good for Villa," he told Marca. "Passing Raul is not easy. This is a generation of players that is breaking records, which is great."
While Villa continues to thrive, though, it was another game without a goal for Fernando Torres. The striker has yet to score since joining Chelsea in January, and his drought extended during a 45-minute spell against Czech Republic, but Del Bosque offered him support.
"We're excited to see how he evolves with Chelsea," he said. "We're there to work to help him."