Valencia came into Sunday's (AEDT) clash having beaten Barca in just one of their past 10 meetings in all competitions, winning last season's Copa del Rey final, while their most recent league success at the Mestalla came in February 2007.
But the Blaugrana were on the back foot from the outset this time around and, while Marc-Andre ter Stegen saved a first-half spot-kick, Gomez made amends as his strike shortly after the restart went in off Jordi Alba for an own goal.
Setien's side belatedly rallied, but after scraping past Granada and Ibiza in the coach's first two games in charge, their luck ran out, Gomez getting a goal himself with 13 minutes remaining.
Real Madrid, not in action against Real Valladolid until Sunday, now have the opportunity to pull three points clear of their bitter rivals at the top of the table.
Gerard Pique gifted Valencia a huge opportunity to take the lead after 12 minutes, his poor clearance prompting an attack that ended with the Barcelona defender clumsily chopping down Jose Gaya inside the area.
However, Ter Stegen came to his team-mate's rescue, brilliantly springing to his left to save Gomez's penalty.
Ter Stegen remained by far the busier of the two goalkeepers and awkwardly helped Gomez's fierce drive onto the crossbar before parrying Kevin Gameiro's follow-up wide.
Barca did not truly threaten until the opening seconds of the second half when Ansu Fati ran onto Lionel Messi's pass on the edge of the area but shot just wide.
It was Valencia who soon led as the off-colour away side struggled to clear, allowing Gomez to shoot hard and low from the right-hand side of the area, his effort deflecting off Alba to wrong-foot Ter Stegen and loop into the roof of the net.
Messi drilled past the right-hand post and sent a lob narrowly over the top, before Jaume Domenech was equal to a 20-yard free-kick as Barca's spell of dominance came too late to rescue a result.
Instead, Gomez got his goal, providing a clinical finish beyond the rooted Ter Stegen at the end of a Valencia counter-attack.