Daniel Agger's 43rd-minute header came at the end of a first half the home side dominated entirely.
The closest they came to opening the scoring before the goal was when Jonjo Shelvey smashed a drive against the post but a host of other chances went begging.
"The game should have been put to bed well before it was," said Rodgers.
"I thought we were outstanding the first half.
"The tempo, we shared the ball, everyone was passing and moving, the creativity [was good] in the team throughout the game and it was just getting that little bit of luck with the final ball.
"No matter how much you dominate the game, if it's still only 1-0 going into the final 10 minutes it can always offer you a bit of challenge where you have to defend.
"There's no question we should have been out of sight, but the most important thing is it was a terrific performance and another win at Anfield.
"We continue with another three points and moving up the league."