And then ex-Rovers trainee Mike Jones sidefooted home with nine minutes left to condemn League One side Tranmere to their ninth defeat in 11 games.
Barnes made six changes to the side that lost 5-0 at Millwall and one of them gave Rovers an ideal start, Curran's hopeful strike slipping through the hands of Bury goalkeeper Wayne Brown at his near post.
But Bury drew level after a neat flick by Jones put Worrall through and the Shakers midfielder rounded Luke Daniels to tap into an empty net.
Rovers almost regained the lead with 15 minutes left as Ian Goodison headed against the far post from an Alan Mahon corner.
Yet it was Bury that booked their place in the Northern area quarter-finals as Jones converted from Tom Elliott's cut-back.