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.
5. Jordan Storey - 7
Marshalling from the middle of the back three, he looked strong in the air and assured on the deck. Missed a great chance from a corner - not for the first time. Defended excellently. Photo: SWFC
6. Liam Palmer - 7
Solid enough, bursting though the middle of the field early doors, but lost possession once or twice when put under pressure. Experience told as the game went on. Photo: Steve Ellis
7. Marvin Johnson - 8
Again offered a constant outlet and after a quiet first half came alive in the second. Put in an excellent cross for Paterson to twice head for the equaliser - and then did the same for Bannan's win-sealer. A confident footballer at current and frankly you love to see it. Photo: Steve Ellis
8. George Byers - 7
Moved the ball nicely in the first half in particular and thundered into one or two tackles, one of which earned him a yellow card. Was involved with giving away the Rovers penalty. Continues his solid form. Photo: Steve Ellis