The result lifted Watford two places to 18th on eight points from 12 games while Norwich, who last won in September when they upset champions Manchester City 3-2 at Carrow Road, dropped to the foot of the table on seven points.
Watford manager Quique Sanchez Flores acknowledged the early goal was crucial and praised the London club's long-suffering fans for their unwavering support.
"You score early and the team becomes more strong and more solid," he told Sky Sports. "I think for us the first feeling is that it is something special for the fans. We wanted to dedicate it to the fans because they have supported for a long time."
Deulofeu fired the visitors into a second-minute lead when he robbed Emiliano Buendia of the ball at the halfway line, cut inside a defender and drove a swerving low shot past Tim Krul.
Gray, who came on for the injured Roberto Pereyra on the half-hour mark, had a shot blocked in the 33rd minute before Bouendia blazed his effort high and wide at the other end.
Spaniard Deulofeu then turned provider in the 52nd when he got a cross in at the second attempt and Gray's deflected back-heel beat Krul at the near post to double Watford's advantage.
The visitors had defender Christian Kabasele sent off midway through the second half for a second bookable foul but always looked more likely to score a third on the break as lacklustre Norwich failed to create any chances.
Norwich's German manager Daniel Farke was furious with his team's sloppy approach and lack of focus.
"If you start such an important game with such a nursery mistake it doesn't help the confidence or the mood," he said.
"We are disappointed, annoyed. We are playing in the best league in the world and you can't make these mistakes."