However, Kirkland conceded a penalty just moments later for a foul on Dunn and it was Rhodes who atoned for his earlier mistake with a calm finish from 12 yards.
Johnson equalised in the 20th minute with a fierce shot following a weaving run through the Blackburn defence.
Wednesday played with more confidence following Johnson's goal and were awarded their own penalty when Reda Johnson was fouled by Josh Morris.
Leroy Lita stepped up from 12 yards to smash home his side's second to edge the hosts ahead.
The hosts sat back in the second half, inviting pressure from a desperate Blackburn side.
Josh King was sent on as Rovers searched for an equaliser and his introduction had an immediate impact as his dangerous cross was headed in by an unmarked Scott Dann.
However, it was the home side who snatched all three points in the 78th minute as Johnson unleashed a shot from 25 yards to score his second of the game and give Wednesday a big victory in their fight against relegation.