James McCarthy opened the scoring for the visitors on the stroke of half-time before Watford defender Jose Holebas equalised from a corner just 38 seconds later.
Former Tottenham goalkeeper Gomes then played a pivotal role in securing a point for the Hornets following a string of fine second-half saves.
“I’m pleased to help the team, but the most important thing is how we approached the game,” Gomes said. “I believe in the game before there was a little bit of a lack of confidence and desire, but we did things differently here.
“Unfortunately, we didn’t come out with the three points, but against a good side like Everton a point is a great result.”
Watford started their return to the top flight in fine fettle, but their form since the turn of year has produced just two victories from 13 matches.
While they will face Crystal Palace for a place in the FA Cup final later this month, they have not won a game in the league for nearly two months.
Gomes added: “We had a meeting during the week, the manager reminded us how we played the first half of the season, how is our game, how is our passion when we approach the game, and we needed to do more of that. We lost that in the last few games but we came again here and we showed the character that we can continue to do that.
Watford have slipped to 15th, with Everton one place and one point better off in the table. McCarthy looked to have secured a crucial lead for the Toffees at the break until Holebas headed home a corner in the third minute of first-half stoppage-time.
“It was important to go to half-time with a lead, then you can refocus and reset,” said Everton boss Roberto Martinez. “What is more disappointing is the nature of the goal. It is a dead-ball situation and it is a header that bounces off the back of a Watford player and ends up in the back of the net. That is the nature of bad fortune at times.
“I was delighted with the way we started again, we were not affected by it and we got in really good situations.
“On another day, with a little bit more momentum, or things going your way, we probably should have taken those good opportunities.”
Gomes, Paredes, Cathcart, Britos, Holebas, Behrami (Guedioura 46), Capoue (Mario Suarez 82), Watson, Jurado (Amrabat 59), Deeney, Ighalo.
Nyom, Prodl, Pantilimon, Abdi.
Robles, Coleman, Stones, Jagielka (Funes Mori 90), Baines, McCarthy, Barry, Deulofeu (Mirallas 78), Barkley (Cleverley 65), Lennon, Lukaku.
Oviedo, Kone, Niasse, Howard.
Kevin Friend (Leicestershire).