Tottenham Hotspur had moved above Liverpool earlier in the afternoon, but Jurgen Klopp's side overcame the league's basement side in impressive fashion to return to the top four.
Liverpool struggled to get going in the first 45 minutes before running riot in the second against a Swansea team that were again under the leadership of Leon Britton.
After a low-key first five minutes, Philippe Coutinho sent Liverpool ahead in the sixth minute when he picked out the top corner with a brilliant effort from outside the Swansea box.
Jordan Ayew had a half-chance to level the scores two minutes later after rising to meet a cross at the far post, but the attacker's header was saved by Simon Mignolet, who kept his spot in the Liverpool XI despite being heavily criticised for his performance at Arsenal on Friday night.
Andrew Robertson was next to come close for Liverpool in the 11th minute as the left-back sent one just wide of the post following smart work from Coutinho once again.
Oliver McBurnie, who was starting his first Premier League game of the season, had an opening for Swansea in the 20th minute after breaking into the Liverpool box, but the 21-year-old's effort was straight at Mignolet.
Emre Can then had a half-chance down the other end, but the German curled the ball straight into the arms of Swansea goalkeeper Lukasz Fabianski, before Alex Oxlade-Chamberlain struck over in the 27th minute.
Liverpool should have doubled their lead in the closing moments of the first half when Roberto Firmino took advantage of a poor pass from Roque Mesa, but the Brazilian fired narrowly wide of the Swansea post after breaking into the box.
Mohamed Salah missed a decent chance for the hosts early in the second period, but Liverpool had their second in the 52nd minute when Firmino volleyed a Coutinho free kick into the back of the net from close range.
Salah twice came close in the five-minute spell of pressure which followed, but the Egypt international fired wide before being denied by Fabianski as Swansea continued to struggle.
Leroy Fer might have halved the deficit just past the hour after finding space inside the Liverpool box, but the midfielder could not hit the target as Swansea passed up arguably their best chance of the match.
It got worse for the visitors in the 65th minute when Trent Alexander-Arnold poked home from inside the box to register his first ever Premier League goal for a dominant Liverpool team.
Less than one minute later, Liverpool had their fourth as Firmino slid home from close range following a smart pass from Salah, who had broken clear of the Swansea defence.
Both managers made changes in the period that followed as Liverpool turned their attention to Saturday's home game against Leicester City, and Swansea to their trip to Watford on the same afternoon.
Renato Sanches was among those to feature off the Swansea bench as the on-loan Bayern Munich midfielder was given the chance to impress in the final 15 minutes at Anfield.
Liverpool added a fifth in the 83rd minute through Oxlade-Chamberlain, however, before Dominic Solanke passed up a couple of brilliant late chances as the Reds served up an impressive second-half performance at Anfield.
Liverpool (4-3-3): Mignolet; Alexander-Arnold, Matip, Klavan, Robertson (Milner 75'); Can, Wijnaldum, Oxlade-Chamberlain; Salah (Lallana 69'), Firmino (Solanke 68'), Coutinho
Swansea (4-3-3): Fabianski; Naughton, Fernandez, Mawson, Olsson; Carroll, Mesa (Clucas 66'), Fer (Sanches 78'); Routledge, Ayew, McBurnie (Abraham, 62')