Welbeck adjusted himself as the ball reared up following a cross from the right, and the former Manchester United and Arsenal forward met it with an acrobatic effort with his back to goal that flew into the top corner from near the penalty spot.
It clinched a first victory since the restart for fourth-to-last Watford and lifted the team four points clear of the bottom three with four games left. Norwich is now 10 points adrift of safety with the same number of games remaining and could be relegated as early as Saturday, when it plays West Ham.
Watford battled back from conceding a fourth-minute goal to Emiliano Buendia, with Craig Dawson scoring the equalizer from a corner in the 10th.
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