Deservedly trailing at the break to Nicky Maynard's 11th-minute opener, substitute Jack Muldoon and Alex Pattison struck in the second period to set up a home clash with Carlisle United in the next round.
Rather worryingly, the Sulphurites were thoroughly second best during the opening 45 to a Rovers side featuring nine changes from their previous league outing.
Mark Oxley had to make a good early stop with his feet to keep out Maynard's deflected effort, but just seconds later he was beaten by the veteran striker, who turned home Kyle Hayde's right wing cross at the near post.
Tranmere's second-string continued to look the more likely, with Oxley forced to tip Maguire's goal-bound header over his cross-bar before Callum McManaman bent a 20-yard strike just wide.
The same player then drew a save out of Oxley with a close-range effort from a tight angle, while at the other end Town failed to create a single clear goal-scoring opportunity.
The second period was an entirely different affair, Simon Weaver switching from an ineffective 4-3-3 system to his tried-and-tested 4-4-2 at the break and introducing Jack Muldoon from the bench.
And within seven minutes of his arrival, the Harrogate substitute levelled matters as he confidently slotted past Joe Murphy at his near post after being played in behind by Josh Falkingham's cute pass.
With the visitors very much in the ascendancy, Lewis Page took aim from long-range and saw his left-footed blast fly narrowly the wrong side of the upright.
They would however move ahead on 71 minutes when Luke Armstrong dispossessed Tom Davies and ran though on the home goal.
Town's leading marksman attempted to dance around Murphy, and although the Tranmere custodian stopped him in his tracks, Alex Pattison followed up to stab into an empty net.
Falling behind instantly jolted Tranmere into life and although they began to see far more of the ball during the closing stages, they failed to really threaten an equaliser.
That was until the very last seconds of five minutes of stoppage-time when Elliot Nevitt burst through, only to be thwarted by a superb block from the onrushing Oxley.