Gabriel Agbonlahor netted the only goal of the game after eight minutes, ensuring Rodgers remains without a home win over Villa in three attempts as Liverpool manager.
A new-look front three of recent signings Lazar Markovic, Adam Lallana and Mario Balotelli struggled to break down a stubborn Villa defence as Liverpool's attack lacked its usual fluency in the absence of Daniel Sturridge and Raheem Sterling, injured and rested respectively.
"We started a bit slowly in the game," he told Sky Sports. "We probably only started to quicken up our tempo when we went 1-0 behind.
"I think the players gave everything today. It was probably a mixture of us not being able to find that quality in the final third of the field and Aston Villa defending well.
"Once they went 1-0 up they defended tightly with lines together; back four, goalkeeper, midfield - they all defended well together.
"We just couldn't find the final pass. We had enough of the ball but we couldn't create so much."
He added: "It's sometimes what happens in football. Normally we're a team that start very, very fast - our tempo and rhythm of the game is at a high level.
"For some reason we just didn't start like that. And, obviously, when a good team gets in front, it is difficult to break them down.
"I can't question the players in terms of their effort and commitment in the game; that was first class."
