Robinson then denied Antonio once more, but from the resulting corner Lewis Buxton drew Wednesday level.
However, the home side were level for just six minutes, as Craig Conway collected a long ball and fired a 25-yard shot beyond Kirkland.
Two minutes before the break, Michael Keane extended Blackburn's lead to two goals with a simple header from close range.
Rhodes almost put the game beyond the hosts after an hour, but Kirkland denied the striker.
In the 72nd minute, Wednesday reduced to deficit to just one goal when Atdhe Nuhiu rolled the ball home after Robinson had blocked Antonio's initial cross.
In the 95th minute, Nuhiu pounced once more to tap the ball into the net to send the home fans into delirium.