Rodgers paid the price for fielding a weakened team as Chico headed the Swans in front just after the half-hour mark.
A Nathan Dyer finished made it two 18 minutes from time and, even though Luis Suarez, off the bench, pulled one back four minutes later by heading home a Steven Gerrard free-kick, Jonathan de Guzman killed the Reds off in stoppage time.
Elsewhere, Norwich scored twice in the final seven minutes to come from behind and 2-1 at home to Tottenham, who also missed a late penalty, in a League Cup clash on Wednesday.
A strong Tottenham side went in front through Gareth Bale in the 66th minute, but a Jan Vertonghen own goal in the 84th minute levelled the scores.
And the Canaries, struggling in the league, completed the turnaround three minutes later as Simeon Jackson followed up to tap home after fellow substitute Grant Holt’s header was parried.
Clint Dempsey had the chance to make it 2-2 from the spot a minute later, but his low effort was kept out by Mark Bunn in the Norwich goal.