Swansea – winners of the competition two seasons ago – could never relax against resilient Rotherham from the Sky Bet Championship, but Garry Monk’s men were always in control after Gomis’ clinical first-half strike.
The decisive moment came after 22 minutes when Nathan Dyer found Gomis on the edge of the penalty area and the 12-times capped France striker followed up his smart turn with a clinical finish low past Millers’ goalkeeper Scott Loach.
After that Swansea pretty much kept Rotherham at arm’s length and, although it was far from a vintage performance from the Premier League outfit, a third successive victory was never placed under too much threat.
Rotherham fashioned the odd chance but their hopes of forcing extra time were over once and for all when substitute Conor Newton sent his shot skywards in the closing stages.
Swansea handed debuts to new boys Federico Fernandez and Tom Carroll in a side showing 10 changes from the one which beat Burnley on Saturday to make it two wins out of two in the Premier League.
Only weekend scorer Dyer kept his place in Monk’s line-up as summer signings Gomis and Jefferson Montero made their first starts.
The kick-off was delayed by seven minutes because of ticketing issues with long queues outside the ground and the wait was hardly worth it as chances were at a premium in the opening 20 minutes.
Dyer saw his shot blocked but Rotherham gave Swansea little hope of encouragement until Jazz Richards broke past Kirk Broadfoot down the left and squared for Montero to test Loach who blocked well with his legs.
Two minutes later Swansea were ahead when the lively Dyer set up Gomis and the finish was proof of the Frenchman’s pedigree.
Lee Frecklington might have pulled Rotherham level but he was unable to get on top of a bouncing ball in the Swansea box and the ball flew high over Gerhard Tremmel’s goal, but there was another warning for Steve Evans’ side when Gomis broke down the left.
Gomis found the willing Montero in support but the Ecuador winger was comfortably held by Loach and the goalkeeper was soon relieved to see Fernandez divert Josh Sheehan’s fizzing free-kick over the bar.
Swansea introduced Jonjo Shelvey at half-time but Rotherham started the second period well and Kyle Bartley had to back-pedal smartly to stop Alex Revell bearing down on goal.
Ben Pringle then volleyed over and Jordan Bowery forced Tremmel to get down low to his left as Rotherham sensed the tie was still very much alive.
Swansea needed to raise their game and Shelvey stirred the home crowd after clever approach play from Richards and Montero down the left.
The ball was squared into Shelvey’s path 25 yards out and the former Liverpool midfielder rifled a powerful shot goalwards which Loach did well to parry away to safety.
Gomis made way to huge cheers in the final 15 minutes and his replacement Wilfried Bony could have twice wrapped the tie up, his first opportunity blocked by Craig Morgan and then firing over when left unmarked from 12 yards.