In a game of few chances, Bishop put the visitors ahead late in the first half and, despite late pressure from the hosts, they held on for their first ever win over Rovers.
Jack O'Connell made his first-team debut in a Blackburn side with 10 changes from the weekend, while Scunthorpe made two enforced changes from the Swindon defeat.
Rovers should have hit the front in the first minute when Josh King sprinted down the left and pulled the ball back for David Dunn, who prodded wide with the goal gaping.
Ben Marshall forced Sam Slocombe into a save with a curling free-kick, but the League One side went ahead after 33 minutes when Bishop met Gary McSheffrey's corner with a glancing near-post header that flew into the top corner.
Slocombe had to be at his best just after the interval when tipping away Marshall's teasing cross, while substitutes Craig Conway and Rudy Gestede combined after 64 minutes but the latter headed over when well placed.
Gestede had the ball in the net 14 minutes from time but it was disallowed as Luke Varney was penalised for a foul on Slocombe as Scunthorpe condemned Rovers to first-round defeat for the second consecutive year.