Rodgers's side were unable to find a way back into the game following the ninth-minute setback, with Philippe Coutinho's strike which came back off the post the closest that the Reds came to an equaliser.
"The players gave everything but we just lacked that bit of quality in the final ball and you've got to give Aston Villa credit," Rodgers told the club's official website. "They defended well once they got the goal, kept their lines tight and defended with a low block.
"We just couldn't get the final pass - it was really frustrating for us. They defended well; the back four and midfield five were in, and with Agbonlahor on the counter-attack they are always a threat.
"We just started slowly in the game and it's unlike us really because normally we start very quickly. In the two games here, we haven't quite started as fast."
The loss leaves Liverpool in eighth place in the table with six points from four games.