Watford dominated possession in the first half, but the Owls created more chances, resulting in Michail Antonio giving Wednesday the lead.
The home side looked more aggressive after the break as they pushed for an equaliser early in the second half.
Marco Cassetti crossed the ball into the box in the 63rd minute, setting up Forestieri, who struck the ball from close range to level the scores.
Forestieri got his second 10 minutes later, this time picking up the ball from Troy Deeney to completely turn the game around for Watford.
The visitors searched for an equaliser in the final minutes with Kieran Lee sending a shot just high before Leroy Lita forced a save from Jonathan Bond in the Watford goal, but their efforts failed to find the back of the net.
The win lifts Watford up to second in the Championship table, while Wednesday sit two points clear of the relegation zone.