But Arsenal were back ahead on the half hour mark as Saka finished off after being set up by Alexandre Lacazette.
Mikel Arteta's side scored a third on 52 minutes as Gabriel Martinelli finished off a wonderful team move. Saka took a quick throw and Odegaard combined with Lacazette for Martinelli to finish off.
"We are getting better and better, the understanding between the players is getting better, and we should have scored more today," Arteta told BBC Sport after the match.
The Arsenal manager put the win partly down to his team's renewed fighting spirit.
"I see how willing they are every day to train and improve. I think they are enjoying playing together and when some days it is not your best day, you somehow manage to win and that is important.
"Credit to Watford, they are fighting for their lives. They changed their team and it is hard to beat them. We have to be very honest with ourselves and our defensive structure was not good enough today. We allowed some chances and it is something we will work on."
Moussa Sissoko netted a second for Watford with three minutes remaining as he chested down and finished from Hernandez's cross.
Watford did put the ball in the back of Arsenal's net after just 17 seconds through Emmanuel Dennis but the goal was ruled out for offside.
The result moves Arsenal into fourth place while Watford remain 19th -- three points behind 17th place Everton, who have three games in hand.