The Bantams had not enjoyed a win in the competition since December 1999 and their hopes of ending a dismal run of early exits suffered a body blow after just six minutes.
A neat lay-off by hard-working midfielder Stephen Jennings caught City's back four flat-footed and while the home fans waited for the assistant's flag Rovers' leading scorer Chris Greenacre calmly fired his side ahead from 12 yards with his seventh goal of the season.
But the visitors then wasted a series of chances before the game was turned on its head in a controversial three-minute spell.
Full-back Lewis Emanuel snuffed out a chance for Delroy Facey with a vital block before sloppy play by City striker Dean Windass gifted possession to Greenacre.
The Rovers frontman saw his strike well saved by keeper Donovan Ricketts and midfielder Mark Rankine should have done better with a follow-up effort which he hit straight at the City shot-stopper.
After half-an-hour Facey wasted another gilt-edged chance when he headed weakly towards goal after being picked out by Gareth Roberts' excellent cross and the let-offs gave City a route back into the contest.
A goal-bound header by skipper David Wetherall was volleyed off the line by Roberts and three minutes later striker Andy Cooke failed to connect properly with a close-range effort after a superb cross from Emanuel.
Soon after an Owen Morrison cross reached Windass at the far post but his header flew just over the bar.
Jennings then poked a near-post chance wide at the start of the second half and Tranmere had a penalty shout for handball waved away before City struck twice.
Rovers were already arguing over a disputed thrown-in when City midfielder Lee Crooks let fly from 30 yards and his thunderbolt rebounded off the underside of the crossbar.
To Tranmere's dismay the linesman confirmed the ball had crossed the line and their outrage increased in the 59th minute when centre-half Ian Sharps was adjudged to have pushed Windass over in the box.
The City marksman then stepped up to calmly side-foot home his 10th of the season to ensure a trip to Yorkshire rivals Barnsley in the next round.
Tempers frayed in the closing stages and Emanuel and Paul Linwood both went into the book after an ugly confrontation by the corner flag.