Greenacre's double delighted the visitors' travelling army of more than 1,800 fans, but in truth the home side were up against it after midfielder Seve Jennings put Rovers ahead after only seven minutes.
Ronnie Moore's League One outfit started brightly and Bantams keeper Donovan Ricketts was forced to push away a Robbie Stockdale shot in the opening minute.
Bradford winger Omar Daley tested Rovers keeper Danny Coyne with a left-foot drive soon after, but Ricketts had to rescue City again when Shane Sherriff fired in from ten yards.
Jennings opened Rovers' account when he picked up a loose ball and rifled home a low drive from 20 yards and Greenacre should have doubled their tally six minutes later when he headed straight at Ricketts from close range.
Tranmere dominated their League Two opponents after that and it was no surprise when Greenacre put them two-up just before the interval.
Again City were guilty of giving away possession and dangerman Chris Shuker picked out his team-mate with a superb chipped pass which sprung the Bantams' offside trap.
Greenacre kept his cool to drill a right-foot shot beyond Ricketts from 14 yards after 37 minutes and from then it was an uphill struggle for Stuart McCall's men.
Rovers kept up the tempo after the break and it took a vital block from City skipper David Wetherall to deny Antony Kay a sight of goal.
City struggled to break down Tranmere's backline and Greenacre delivered the killer blow 22 minutes from time, when he chased down a long through ball and held off Wetherall to fire in his second from just inside the box.
With a third round spot in the bag Rovers ran down the clock and the only black mark on the day was the sight of skipper Ian Goodison being stretchered off after picking up a leg injury.