Wednesday were the better side, creating a host of chances to score, but a combination of wasteful finishing and an assured performance from opposition keeper Dillon Phillips kept the scoreline blank until Fletcher struck at the death.
Forestieri then tried a low shot from inside the area which Phillips pushed wide.
Forestieri had time to pick his spot but his attempted chip was easily gathered by Phillips and the Wednesday forward was again in the thick of the action, firing in a shot which was deflected wide.
Forestieri had another chance within a minute of the restart when his shot hit the side netting after Barry Bannan slipped the ball through to him on the left side of the area.
It was the start of a good spell for Wednesday with Forestieri again threatening, seeing an effort deflected wide. Kadeem Harris also had a shot saved by Phillips.
Fletcher then posed problems for the visitors, first seeing his effort kept out by Phillips and then having a firm header superbly saved by the keeper. From the resulting corner, Wickham shot wide.
The visitors went close to taking the lead against the run of play when Taylor's long-range strike thumped against the bar.
Forestieri was involved again soon afterwards, trying a shot from distance which was well saved by Phillips.
When Taylor threatened from close range, Wednesday skipper Tom Lees made a vital challenge to block his effort.
With the clock ticking down, Palmer's free-kick was flicked on by Lees and Fletcher headed in from close range with home fans reacting with a mixture of relief and elation.