Simpson earned Walsall their second successive League win in what could be his final appearance for the club.
Simpson has been playing on a non-contract basis and the club have told him they cannot afford to offer him a permanent deal.
He will make a decision on whether to stay at the club this week - and if he does leave he signed off in style.
His low 44th minute drive sentenced Wednesday to their first defeat in eight games.
It was also the third game in a row that Wednesday have failed to score.
But they should really have taken something from the game after an improved second half when substitute Leon Knight sparked the team into life.
Boosted by their return to winning ways at Millwall in their previous game, Walsall made the early running against Wednesday but struggled to find their touch in front of goal until Simpson struck.
Australian international striker David Zdrilic, who had scored three goals in the previous two games, was the chief culprit for Walsall.
He volleyed just wide from 20 yards after 11 minutes but was then guilty of a glaring miss 12 minutes later.
A weak back-pass from Wednesday skipper David Burrows was intercepted by Steve Corica.
The ball was eventually forced into the path of Zdrilic but he could only drag his shot across the face of an open goal.
He then fired over the bar from ten yards after Darren Bazeley and Darren Wrack had combined to create the opening.
His failure to convert any of the chances looked set to be costly until Simpson made Walsall's domination pay on the stroke of half time.
Wednesday only half-cleared a ball into the area from Walsall skipper Martin O'Connor and Simpson collected the clearance before drilling a low 20-yard drive past goalkeeper Kevin Pressman.
While Walsall dominated the opening period, Wednesday posed more of a threat in the second half following the introduction of Knight.
His 55th minute close-range drive tested Walsall goalkeeper James Walker for the first time.
He then created a golden chance for striker Lloyd Owusu with a teasing right-wing cross after 62 minutes.
Owusu was left facing an open goal but failed to even make contact with the ball and it proved to be a crucial miss.