Chico Flores and Nathan Dyer’s goals either side of the break put the visitors in front, before substitute Luis Suárez glanced home Steven Gerrard’s free-kick.
But Jonathan de Guzman added a third in stoppage time to book the Swans’ place in the quarter-finals.
“Swansea were brighter than us. In the first half, we were too slow and ponderous,” said Rodgers.
“We introduced Luis Suárez and Steven Gerrard and that gave us great impetus but we gave away a disappointing second goal.
“Overall, I thought the best team won. We created chances but we have to finish them. There are no complaints from us.”
Leeds United v Chelsea
Swansea v Middlesbrough
Norwich v Aston Villa
Bradford City v Arsenal
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