Things seemed to be going their way when Marc Bridge-Wilkinson curled a magnificent 25-yard free-kick into the top left corner, before veteran keeper Andy Dibble saw it on its way.
An earlier drive from Mark Robertson cleared the bar and Jim Goodwin tested Dibble with a brisk shot from short range.
Dibble nearly gave away an easy goal when he allowed a back pass to roll towards his line then scrambled it away with inches to spare.
Having gone behind, Wrexham tried to catch up, badly needing a win after having 10 points docked for going into administration.
Hector Sam saw a shot hacked off the line and then another effort from Andy Holt was blocked in a crowded goalmouth.
Holt was close again with a firm header from a Simon Spender cross only for Neil Cutler to deny the Wrexham wing-back with a good save.
Cutler came to the rescue again with a brilliant save on 24 minutes, stretching to tip Juan Ayestarán Ugarte's header around the post.
Another header from Dennis Lawrence went wide before Wrexham levelled on the stroke of half time.
The goal came from a good move, with Chris Llewellyn crossing for Holt to slant his angled header in.
It was all Wrexham after the break as Darren Ferguson tested Neil Cutler with a rising shot and Shaun Pejic fired narrowly wide.
The winner came on the hour, again with Ferguson involved, providing a defence splitting cross which Ugarte headed on for Sam to plant a firm header home.
Stockport caretaker manager Mark Lillis was disappointed to lose after taking an early lead.
He said: "We were 1-0 up and we should not have let Wrexham get back in it.
"That goal on half time gave them just the lift they needed to go on and win."
He added: "We need to concentrate a lot more but we showed total commitment and got going right at the end.
"We are trying to get a winning mentality and move away from the losing streak we are in right now.
"The situation should improve when things get sorted out at the club.
"Then we will know where things are going."
Wrexham manager Denis Smith was happy to start on the long climb up the table after having 10 points docked.
"We made hard work of what should've been an easy victory," he said. "We made lots of chances but couldn't put them away."