The Shrews had more than matched their illustrious opponents at New Meadow, and had gone close to breaking the deadlock in the opening half when Shaun Whalley’s looping free-kick found the angle of post and bar.
The home side also threatened from set-pieces as their early intensity proved too much for the Black Cats.
But Omar Beckles’ miscued clearance flew in past Joel Coleman to give Jack Ross’s side the breakthrough, and they capitalised by adding a second through sub Luke O’Nien in the 84th minute to wrap up the win.
Askey said: “To concede that, from an own goal, was a bit of a kick in the teeth when we’d stopped them from playing.
“In the final third we need that bit of extra quality at times when we’re breaking through. It was either the final pass or shot that we didn’t have.
“Overall I was pleased with how we performed. We’d worked hard to stop them from playing and I thought we did that.”