The chances continued to be created and not converted as Sturridge missed the target with a close-range header, and Coutinho, who joined from Inter Milan last month, stabbed the loose ball wide with the goal at his mercy.
Referee Howard Webb was also unconvinced by Kemy Agustien's challenge on Suarez close to the touchline, but fortunately for Liverpool assistant Mike Mularkey saw it differently and immediately flagged for a penalty.
Gerrard, having failed from the spot in a 2-0 loss against West Bromwich Albion on Monday, went for the opposite corner and succeeded in beating Michel Vorm for Liverpool's first goal in the 34th minute.
Sturridge had one more chance before the break but having been put through by Suarez his left-footed toe poke was blocked by the head of Kyle Bartley.
Swansea had shown little threat in the opening 45 minutes and any ideas the visitors 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' defense and beat Vorm was all that was required.
However, a well-worked goal enabled Enrique to poke home Sturridge's pass in the 50th minute, before Suarez went past both Swansea's center backs to fire home the fourth goal six minutes later.
Sturridge scored courtesy of the generosity of Gerrard, who relinquished penalty duties in the 71st when substitute Wayne Routledge handled the ball.