The visitors took a surprise lead on just nine minutes when Abu Ogogo intercepted a poor clearance before firing towards goal.
Blackburn managed to equalise on the half-hour mark after Ryan Nyambe sent a cross from the byline into the path of Nathan Delfouneso, who nodded into the top corner through a Liam Lawrence deflection.
The hosts' lead would last all of one minute, however, as failure to clear the lines allowed Tyrone Barnett to drill a left-footed volley past Steele from the edge of the area and put the Shrews ahead again.
Shrewsbury pushed forward and played attacking football after the restart, as Collins blazed over the bar from close range in the 54th minute and Martin Woods followed suit midway through the second half.
Koita surfaced again in the dying stages of the game as he smashed a header narrowly wide of the far post, but he could not prevent his side from succumbing to defeat and an early exit from the cup against League One opposition.