The 20-year-old scored the important second moments after the break following Steven Gerrard's first-half penalty.
Jose Enrique, Luis Suarez and Daniel Sturridge, also from the spot, finished off the visitors who rarely looked interested in, or capable of, making a contest of it.
Liverpool started well as England right-back Glen Johnson planted a free header well wide, Gerrard drilled a free-kick into the wall and Suarez fired the rebound over, before the Uruguay international whipped a 25-yard free-kick just over the bar, where he also dispatched a dipping volley.
Howard Webb waved away Sturridge's penalty appeal and was also unconvinced by Kemy Agustien's unnecessary challenge on Suarez close to the touchline but, fortunately for the home side, his assistant Mike Mularkey saw it differently and immediately flagged for a penalty.
Gerrard, having failed from the spot in front of the Kop against West Brom, went for the opposite corner and succeeded in beating Michel Vorm for Liverpool's first goal.
Swansea had shown little threat in the opening 45 minutes and any ideas they may have had about staging a comeback were snuffed out within 10 minutes of the restart. In truth, the 16 seconds it took the 20-year-old Coutinho to skip through the visitors' defence and beat Vorm was all that was required.
However, a well-worked goal saw Enrique poke home Sturridge's pass before Suarez went past both Swansea's centre-backs to fire home the fourth.
The persevering Sturridge was denied again by Vorm as the Reds ploughed forward almost at will with Swansea seemingly powerless to stop them or provide any effective response of their own.
He finally got his goal but only courtesy of the generosity of Gerrard, who relinquished penalty duties when substitute Wayne Routledge handled.