The Merseysiders had won the competition last season under Kenny Dalglish to end a six-year trophy drought but the Scot was then sacked in the summer and Rodgers was recruited from Swansea.
It was speculated whether the Welsh club would struggle without their former boss, who guided them to 11th place in the Premier League in their debut campaign after winning the Championship play-off final in the previous term.
But, despite a little wobble, Michael Laudrup has continued much of the good work and a win at Anfield, which came through Chico, Nathan Dyer and Jonathan de Guzman, while Luis Suarez headed a consolation, and a Capital One Cup quarter-final tie at home to Middlesbrough is another positive.
Liverpool dominated the opening 10 minutes but failed to test Swansea goalkeeper Gerhard Tremmel, with Joe Cole, who was starting his first game since August among nine changes from the Merseyside derby, seeing a scuffed long-range shot deflect for a corner.
Swansea posed their first threat after quarter-of-an-hour but Dyer failed to control when behind the home defence after Jamie Carragher had misjudged a long ball.
Laudrup had made just four changes to the team who started last weekend’s defeat by Manchester City and it showed in a game which followed a similar pattern to that at the Etihad Stadium, where Swansea’s confidence grew as the first half progressed.
As the half-hour mark arrived, Liverpool were having to chase the ball and were surrendering territory before being fortunate to see De Guzman chip over the crossbar after the hosts had been caught on the counter-attack.
The warnings had been there for Liverpool and so it was no surprise Rodgers’ team fell behind in the 34th minute when Chico, Laudrup’s £2million July signing from Real Mallorca, headed his first goal for his club from De Guzman’s right-sided corner.
Having lacked a cutting edge in the first 45 minutes, Liverpool sent on star men Suarez and captain Steven Gerrard at half-time as substitute replacements for ineffective teenager Samed Yesil and Cole.
The changes gave Rodgers’ men a greater presence and Gerrard saw Tremmel tip a 25-yard effort on to a post before Stewart Downing made a hash of the follow-up in a mess which epitomised the £20million former Aston Villa winger’s troubled Anfield career.