The forward struck in the second minute at Carrow Road before teeing up Andre Gray to double the lead early in the second half.
Christian Kabasele was sent off 65 minutes in, but Watford held firm to move above Norwich and Southampton into 18th place on the table.
Deulofeu scored after only 76 seconds, the second-fastest Premier League goal Watford, when he stole possession from Emiliano Buendia in the Norwich half, tricked his way into space and slotted the ball past Tim Krul.
Norwich was largely blunt in attack and found itself further behind 52 minutes in when Gray's attempted backheel from Deulofeu's cross went in via a big deflection off Jamal Lewis.
Kabasele earned a second yellow card when he wrestled Josip Drmic out of play on the touchline, but Quique Sanchez Flores's men were not unduly troubled as they saw out a first league win since a 2-1 victory away to Huddersfield Town in April.