Javier Hernandez managed to break free from the attention of Jose Enrique to slot home Wayne Rooney’s corner early in the second half, ending Liverpool’s run in the league cup for this season.
Rodgers was happy with some parts of the game but admitted it was a very sloppy goal to concede.
"It was a very close game and to decide it on a corner at the beginning of the second half was disappointing,” he told Sky Sports 1.
"The players have given it everything. I thought they played it really well.
"In the second half we had numbers going forward, we created opportunities but couldn't make the final pass count.
"It was very small margins, I'm very disappointed, it was a really loose goal."