Jordan Rhodes put the visitors ahead from the spot but the home side hit the front thanks to a Johnson strike and a Leroy Lita penalty nine minutes before the break.
Scott Dann levelled for Rovers after 71 minutes, before Johnson put his side back in front seven minutes later with his second impressive strike.
Johnson's moments of individual brilliance were the highlights of a poor game in South Yorkshire, although the win moved Wednesday three points clear of the drop zone.
Wednesday forced a succession of early corners but it was the visitors who created the first chance with David Dunn's accurate cross forcing a scrambling Chris Kirkland to deflect the ball to safety.
After 12 minutes Kirkland was scrambling again, but this time he was caught out.
Miguel Llera was comfortably beaten for pace by Rhodes' who got on the end of a long ball over the top. Although Kirkland stopped the Scottish striker, the ball fell to Dunn who went down under the stopper's challenge.
Rhodes sent Kirkland the wrong way for his 25th goal of the season.
Within eight minutes Wednesday were level. A ball into the box fell kindly for Johnson, who cut inside and with Rovers defenders slow to react, fired hard and low into the corner of the net.
The hosts should have taken the lead when Lita's dummy played in Jeremy Helan and with Blackburn ball-watching he turned and screwed his shot over.
Blackburn were dropping deeper and deeper and after 35 minutes found themselves in trouble.
Reda Johnson was bundled over as Grzegorz Sandomierski claimed a cross into the box and linesman Mathew Buonassisi immediately flagged for a penalty.
There was little protest apart from the goalkeeper and Lita tucked away the spot-kick for his fourth Wednesday goal.
After a lacklustre opening 10 minutes of the second half, Gary Bowyer brought on Kazim-Richards for Cameron Stewart, while Kieran Lee replaced Stuart Holden.
Josh King came on for Lee Williamson as Blackburn again tried to increase their attacking options, but with little creativity and Wednesday reasonably strong at the back, chances were hard to come by.
With Wednesday insisting on playing long balls forward, Howard for Lita was a logical substitution after 70 minutes.
However it was at the other end where Wednesday were starting to have trouble and seconds later Blackburn were level, when King's inswinging cross found Dann who got goal-side of Llera to head home.
Their lead was not to last long. Johnson broke forward and with Blackburn again retreating, fired an unstoppable piledriver into the bottom corner.