Swansea manager Michael Laudrup rested a host of regular starters with next weekend's Capital One (League) Cup final against fourth-tier Bradford City in mind, a decision which spectacularly backfired.
Steven Gerrard's penalty on 34 minutes set Liverpool on their way, the midfielder making no mistake having had a spot-kick saved in the home defeat by West Bromwich Albion on Monday.
Coutinho, who joined last month from Inter Milan on a long-term contract, ran at the Swansea defence and beat keeper Michel Vorm soon after the restart to make it 2-0 as Liverpool quickly settled the game in a rampant spell.
Coutinho was also involved in the build-up in Liverpool's third goal four minutes later, linking up with Luis Suarez before Daniel Sturridge crossed for Jose Enrique to prod in.
Suarez hit a fourth on 56 after twisting and turning in the area before Sturridge made it 5-0 with a second penalty.
It was Liverpool's first victory over a team in the top half of the Premier League this season and a first for manager Brendan Rodgers over his former side.