Adams picked up just one point in five games as he failed to prevent City from being relegated from the Premier League after replacing Chris Hughton during the closing stages of last season.
His side also lost their Sky Bet Championship opener at Wolves on Sunday, but they got their new campaign up and running at Carrow Road with a well-deserved victory on Saturday.
Watford were struggling from the moment Swedish defender Joel Ekstrand saw red in the second minute and Norwich went on to dominate, with well-taken goals from Bradley Johnson, Lewis Grabban and Alex Tettey settling it.
The match got off to a dramatic start with a smart save, a red card and a header just over the bar all in the space of the opening two minutes.
John Ruddy did well to gather Troy Deeney's low free-kick before Ekstrand appeared to catch Nathan Redmond with a flying elbow as the pair challenged for the City goalkeeper's clearance.
Referee Paul Tierney decided it was dangerous play and sent the Watford man off, with Michael Turner then nodding Wes Hoolahan's deep cross just over.
The early excitement proved short-lived however, with the Hornets seemingly happy to soak up the pressure as the half progressed, although Redmond went close with a right foot strike from just outside the box as the hosts continued to press.
Norwich finally got the goal their dominance merited on 32 minutes as some neat approach play from Hoolahan had the visitors backpedalling. The ball fell nicely for Kyle Lafferty who cut it back for Johnson just outside the box and the midfielder skilfully curled the ball over keeper Heurelho Gomes, who was well off his line, and into the back of the net.
As the half drew to a close, Watford were struggling to get out of their own half, and Gomes made a good block to prevent Grabban from stretching the Canaries' advantage.
Watford introduced Lloyd Dyer at the break in an attempt to give their side more of an attacking edge and it took the winger only nine minutes to draw an excellent save out of Ruddy following a quick break down the right.
Norwich continued to look the more dangerous side however and Lafferty should have made it two on 56 minutes after being put in by the excellent Hoolahan.
But the home supporters need not have worried, with two goals in as many minutes effectively settling the game as a contest.
Both were impressive long range efforts, with summer signing Grabban setting the ball rolling on 59 minutes with a delightful chip over Gomes for his first Norwich goal and Tettey then blasting home a low shot in City's very next attack.
The game petered out to a quiet conclusion after that, although the appearance of Leroy Fer from the bench for his first action since playing in the World Cup for Holland gave the home fans a further boost.