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.
9. Massimo Luongo - 7
Started his a fourth league match in a row for the first time since November/December 2020. Contributed to giving away Donny's penalty with Byers. Moved the ball nicely in the main and offers assurity in that midfield. Photo: Steve Ellis
10. Barry Bannan - 7
Was off-balance to shank an early chance wide and made some nice touches, but struggled to pick the lock of a determined Donny defence. Had a poor penalty saved by Mitchell but turned Johnson's cross home late to seal it. And what a celebration. Photo: Steve Ellis
11. Florian Kamberi - 4
Passive. Barely registered a meaningful touch in the 45 minutes he played and when it did it was most often a tackle. Hooked at half-time. Photo: Steve Ellis
12. Sylla Sow - 4
Failed in what felt like an audition piece. Like Kamberi, he was too passive and wasted the rare opportunities he got got in ahead of the Donny defence. Headed tamely at Rovers keeper Mitchell with Wednesday first real chance of the match. Hooked at half-time. Photo: Steve Ellis