A second from Sturridge, who now has four goals in three appearances this season, and a Jordan Henderson strike eventually put Liverpool through, but Rodgers has admitted that the Magpies impressed him with their ability to respond to their early deficit.
He told Sky Sports News: "I think you have to give a big credit to Notts County because at 2-0 down here at Anfield, we're in good form at the minute. It was a great fightback for them and that will bode well for their season because they showed great character and quality.
"We've seen it in football before - 2-0 is always a dangerous scoreline no matter who you're against. We're obviously disappointed with the goals we conceded and our management of the game, but across the board, we're glad to get through."
Notts have picked up just one point from their opening four League One fixtures.