Bernard’s first-half goal was enough to give Everton their first win of the season and hand Watford a fifth straight Premier League defeat.
The Hornets lost their last three matches of last season, and have yet to score in 2019-20.
They came closest to equalising when Craig Dawson’s header hit the bar, following Bernard’s shot that beat keeper Ben Foster at his near post.
Watford improved after the break, but Abdoulaye Doucoure missed late on.
Troy Deeney had been denied by a fine save from Jordan Pickford earlier in the second half, when the Watford captain went one-on-one with the Everton keeper.
And a Deeney shot that deflected off Michael Keane set up Doucoure inside the Everton area only for the midfielder to pull his shot wide of the target.
Everton striker Richarlison had two good chances to score against his former club with headers but missed both, while substitute Moise Kean rolled a shot just wide of Foster’s right post in stoppage time at the end of the match.
The Toffees have not conceded a goal in six games at home since their 0-2 defeat by Manchester City on 6 February.
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