After first-choice goalkeeper Petr Cech injured a hamstring at the end of the first half, Leno was forced into action for the remainder of the game.
The German star made two crucial saves to keep Watford from scoring for the first time in the league this season.
The Arsenal signing was thrilled with his Premier League debut against the Hornets last weekend.
“It was a difficult start when you don’t think you’re going to play and then Petr is injured – I hope he will get well soon,” Leno told Arsenal’s website.
“It was a good game for me with a difficult start, but I think that I made some good saves and we kept a clean sheet, so I’m happy with my first game in the Premier League.”
Leno joined Arsenal from Bayer Leverkusen in the summer for a figure reported to be as high as £22.5m.
He began playing professional football for VfB Stuttgart’s second team before moving to Leverkusen in 2011.
The German star made 233 league appearances for Leverkusen over seven years before moving to Arsenal.
His move was initially met with skepticism from former Arsenal players and Leno will be relieved to have had the perfect debut.
And it looks like he will be called into action in the weeks ahead as Cech has been ruled out for at least two weeks.
Arsenal face Qarabag and Sporting Lisbon in the Europa League in that time, as well as a trip to Fulham and hosting Leicester in the Premier League.
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