A poor second half devoid of ideas saw the Owls head into the break behind to a penalty, but a determined last half hour earned Wednesday the win.
Goals from Callum Paterson, Saido Berahino and Barry Bannan won it, with Bannan having earlier missed a penalty.
Darren Moore must surely take great praise for three telling substitutions.
Here are our ratings from a freezing cold day at the Eco Power Stadium.
13. Callum Paterson - 8
Made a difference coming on t half-time and offered something to aim at. Worked hard. Glanced a header over the bar - a good chance - on the hour. Won the penalty. Changed the game. Photo: Steve Ellis
14. Saido Berahino - 8
Shot into the side-netting shortly after coming on. Looked busy and produced some nice touches for a player with such little game time of late - and what an excellent finish for his first since September. Photo: SWFC
15. Lewis Gibson - 8
On for Brennan to readdress the balance of the defence - he did that and more, offering an assist for Berahino's finish. Photo: Steve Ellis