The Anfield outfit allowed Chicharito to score the only goal of the game, though the Northern Irish boss was pleased with his side's efforts, especially the returning Luis Suarez
Rodgers named a strong starting line-up for the third-round clash at Old Trafford on Wednesday, including Luis Suarez after the expiration of the striker's 10-match ban.
A lapse in concentration when defending a corner just after half-time allowed an unmarked Chicharito to score the only goal of the game, and, despite being pleased with his side, Rodgers rued the manner in which they conceded.
"It was a very close game - decided on a corner at the beginning of the half that we're disappointed with," the Liverpool boss told Sky Sports.
"It just drops into the six-yard box and it's very disappointing to say the least.
"The players played really well. They had a few days to adapt to a new system. They created some opportunities but couldn't make the final pass count and didn't work the keeper enough.
"I can have no complaints. One lapse of concentration has cost us."
Suarez came close to getting Liverpool back into the game when he hit the crossbar with a free kick in the second half, and Rodgers praised the 26-year-old's performance following his return from suspension.
"I thought Luis Suarez was excellent," he added. "You could tell he'd been out for a long time as he tired at the end.
"But he's obviously a threat. I think adrenaline gets you over the line in the first game. He's a fighter. He's so determined and will get ready for the weekend."