The Reds had Jose Enrique's first-half strike ruled out for a marginal offside decision, while teenage prodigy Raheem Sterling rattled the bar and Jonjo Shelvey's late strike was beaten out by Gerhard Tremmel.
Man of the match Pablo Hernandez had Swansea's best chances. The Spain winger curled narrowly wide in the first half before having his low-free-kick brilliantly tipped wide by Jose Reina.
Ashley Williams, who shook hands with Luis Suarez prior to kick-off after his criticism of the Liverpool striker, also saw former Swan Joe Allen clear his header off the line.
Despite seeing his side only secure a point, which extended their unbeaten league run to eight games but keeps them below Swansea in the table, Rodgers will have taken heart from another promising performance as he looks to rebuild at Anfield, with Sterling, Enrique and Glen Johnson excelling.
And with his intimate knowledge of Swansea's style, the Northern Irishman, who received a good reception from the home faithful, set Liverpool up to press his former club as high up the pitch as possible.
It proved effective, and Johnson should have given them the lead from Steven Gerrard's corner after Allen had dispossessed Leon Britton.
Wayne Routledge forced a sharp near post save from Reina after turning on the edge of the area, but the game was being played almost exclusively in the Swansea half.
Suarez should have capitalised when Swansea again surrendered the ball, but strayed offside as Enrique advanced, before the Spaniard steered narrowly wide from Sterling's superb cross.
Enrique turned provider with a lovely pass for Johnson in the 26th minute, but Tremmel was out quickly to block his effort.
Swansea were limited to counter-attacks, and one sharp break saw Hernandez go close to breaking the deadlock, his curling strike beating Reina but also the far post.
Liverpool went straight up the other end and failed to convert a trio of clear-cut chances that would have given them a deserved reward for their dominance.
Angel Rangel's brave block denied Suarez a simple finish, before Sterling struck the bar with a thunderous volley.
Enrique had the ball in the net after 34 minutes from Suarez's delicate pass, but the assistant referee ruled offside, although replays suggested he may have been level.
Hernandez, Swansea's most lively operator, shot over the bar as half-time closed in, but there was no denying Liverpool's superiority.
Michael Laudrup brought Ki Sung-yueng on for the anonymous Itay Shechter at half-time and pushed Michu up front in an effort to alter the pattern of the game.
The change succeeded in solidifying the Swans midfield but the visitors still looked the more likely to break the deadlock, with Tremmel beating away a fierce angled drive from Suarez.
After that scare the hosts enjoyed their best spell of the game.
Michu had a header saved by Reina when Hernandez broke free of Johnson's marking before Allen denied his former team by clearing a Williams header off the line at the far post.
Nathan Dyer, just on as a substitute, raced through on goal after robbing Stewart Downing at left-back, but could not beat Reina as the two slid for the loose ball and collided. Both players were fine to continue after treatment.
Michu had a penalty appeal turned down after a grapple with Daniel Agger, and Swansea would have gone behind had Liverpool not blown a golden opportunity with eight minutes to play.
Suarez and Sterling broke from a Swansea corner and had only Ben Davies in front of them, but a laboured pass from the winger forced the Uruguayan wide and Tremmel held his shot.
Hernandez thought he had stolen the points with just three minutes remaining, only for Reina to make a superb fingertip save, before Tremmel blocked Jonjo Shelvey's fizzing strike to ensure the points were shared.
Source: PA