Reading, in their third match under the Portuguese boss, paid for slack defending in the first half and trailed 3-0 at the break.
Swansea were ruthless, with Oliver McBurnie, Connor Roberts and Mike Van Der Hoorn all on target before half-time.
The visitors added a fourth early in the second period when McBurnie converted a penalty.
Although the hosts reduced the gap in the 77th minute through Callum Harriott, it was little more than a consolation and they are now without a win in 10 matches.
Gomes, who replaced the sacked Paul Clement last month, had previously overseen a 1-0 defeat at Millwall and, last time out, a 0-0 draw at Queens Park Rangers.
Relegation-threatened Reading had not won in nine matches and had scored only one goal in five outings.
Swansea's play-off hopes – and a swift return to the Premier League – had seemed to fade after a run of three games without a victory and five defeats in eight fixtures, but a return to winning ways left them seven points adrift of the top six.
Gomes named the same starting XI from Loftus Road but his side were rocked in only the second minute.
Full-back Kyle Naughton crossed from the right and the unmarked McBurnie comfortably nodded in his ninth goal of the season.
Reading almost produced a quick response but central defender Tiago Ilori heading a chance over the crossbar.
John Swift then sent in a powerful 30-yard free-kick, which goalkeeper Erwin Mulder did well to parry to safety.
Sone Aluko cut in from the right, only to blaze a shot over the bar, and Yakou Meite nodded wide from a Swift free-kick.
But Swansea doubled their lead on the half-hour mark when left-back Roberts stole in unattended at the far post to poke home Leroy Fer's deflected cross.
And the Welsh visitors made it 3-0 on the stroke of half time, when Van Der Hoorn guided in Matt Grimes' free-kick to the far post.
If Gomes was hoping for a Reading rally in the second half, he was disappointed.
Within three minutes, Swansea had extended their lead after Garath McCleary clumsily fouled Wayne Routledge in the home area. McBurnie stepped up confidently to thump home the resultant penalty.
Reading pegged a goal back 13 minutes from time, when Harriott exchanged passes with fellow substitute Danny Loader before executing a neat finish.
But by then Swansea had done enough to start 2019 with a victory.