The 19-year-old striker, on loan from Sunderland, crashed home a volley from close range in the 70th minute as Wednesday secured only their second win in their last six league matches.
The Foxes, who needed a stoppage-time header from on-loan Manchester United defender Michael Keane to salvage a point against Leeds in midweek, have now won only one of their last seven league matches.
Keane almost repeated the feat in the 81st minute, but this time his header from substitute Anthony Knockaert's corner hit the post.
Leicester striker David Nugent forced a superb early save from Owls goalkeeper Chris Kirkland with a dipping 20-yard shot and Keane headed Marshall's corner over the crossbar.
Nugent rifled another effort from outside the area wide before Wednesday responded with their first chance in the 12th minute, Leroy Lita flashing a header wide from Michail Antonio's cross.
Marshall provided a telling cross, this time for Harry Kane shortly after the restart, but the on-loan Tottenham forward headed tamely off target.
Owls defender Reda Johnson had to go off injured and was replaced by Wickham just before the hour mark and Foxes manager Nigel Pearson sent on Knockaert for Kane soon after.
Marshall blazed shot from 12 yards over the crossbar and Leicester were then punished for their missed chances.
Lewis Buxton swung a cross over from the right and Wickham crashed the ball into the roof of the net, on the volley, from eight yards.
Wednesday held on without any further near-misses to send the Foxes spinning to a third defeat in their last five home league matches.