The away success arrived despite a spirited Saints performance in the second-round tie.
Richard McGivern produced an early save before Rovers carved out openings for Cecil Corbett and Jonny Black, the latter effort clipping the crossbar off Jamie Uprichard’s cross.
Black made his mark with a crisp drive to cap Corbett’s industry from the left flank.
Corbett then turned from provider to scorer by slotting home AJ Gallagher’s pass at the end of strong defensive play from Gary Whiteside.
Black’s shot clipped the upright but the striker added to Rovers’ tally by collecting a Gallagher cross, controlling the delivery and smashing home a shot off the underside of the crossbar.
Gallagher failed to make the most of a promising opening but St Mary’s managed to maintain a threat thanks to Feeney’s pace.
Chris Mackle opened his account in Rovers colours against former club St Mary’s for the afternoon’s fourth and final goal.
Whiteside found Black and the striker’s pass released Mackle to race clear and convert.
Uprichard was named the leading player for Rovers.
TANDRAGEE ROVERS: Richard McGivern, Jamie Uprichard, Chris Mackle, Colin Bingham, Gary Whiteside, Lee Wells, Shea Skelton, Jonny Black, Cecil Corbett, AJ Gallagher. Subs: Simon McClean, Scott Walker, Ricky Dobbs.
“It was a funny game as we started well but Tandragee Rovers deserved the overall win,” said St Mary’s Gerry Fay. “Rovers have plenty of experience and we failed to take our chances.
“Ruairi Feeney came close and Stephen Cusack had a couple of efforts.”