The holders were dumped out of the competition in a 3-1 fourth round defeat at Anfield as their manager suffered an unhappy first reunion with his former club, who he left in the summer.
Rodgers confessed his team, which included nine changes to last weekend's Merseyside derby, paid the price for a slow start when falling two goals behind to Chico's first-half header and Nathan Dyer's second-half goal.
Liverpool had improved after half-time, when Steven Gerrard and Luiz Suarez, who scored his side's goal, arrived as substitutes but Swansea still progressed to the quarter-finals as Jonathan de Guzman made sure of the win in injury time.
And Rodgers said: "I thought the best team won. I thought we were slow in the first half. It took the introduction of Steven Gerrard and Luis Suarez.
"Obviously we made a lot of changes but I just felt the tempo and the intensity of our game was way too low.
"It is no surprise, I know Swansea inside out, so I know the brightness and quality they have."
Liverpool next host Newcastle United in the Premier League on Sunday before an away date with Anzhi Makhachkala in the Europa League on the following Thursday and then a top-flight weekend trip to Chelsea.
And Rodgers said of his decision to rotate his squad against Swansea: "We have a lot of games and I have to use the squad. It is as simple as that.
"Otherwise I will have no players left by the time we get to Christmas."