Goals were scored either side of half time by firstly Chris Shuker and then substitute Steve Jennings, as Ronnie Moore's side never looked like losing.
The manager made three changes from the side that was deafeated by Port Vale on Friday, as John Achterberg, Danny Harrison and Jason McAteer all started instead of Gavin Ward, John Paul and Paul McLaren.
The first twenty minutes of the first half were increadibly dull, as the huge attendance of 1093 (of which around 300 were Tranmere fans) failed to create any sort of atmosphere - bar the slow hand clap.
Niether side had many chances and nor did either look threatening when attacking, as the likes of Chris McCready for Tranmere and Dave Challinor for Bury stopped anything coming their way.
Rovers had the first chance of the game, from a free-kick, after Steve Davies had been fouled on the edge of the penalty area. Chris Shuker took the free-kick and placed the ball into the top right-hand corner, leaving the keeper stranded.
This was the only real action of the first half, although John Achterberg was kept busy by calling defender Ian Goodison "Zola" and he made one or two good claims from crosses, including one cat like spring to collect a ball on the edge of the six yard box.
The second half was alot more lively, as the Rovers fans began to get into motion with a few chants, although the traditional "Super White Army" was only chanted once through-out the night.
On 49-minutes, Nicky Adam's shot-come-cross zipped across the wet turf, eluding everybody on it's way out for a throw-in, whilst Flitcroft hit a shot wide.
Danny Harrison found Kevin Ellison in the Bury penalty area no 59-minutes, but the winger's weak shot was easily saved by on-loan keeper Edge, whilst Chris McCready and Steve Davies both put shots wide for The Super White Army.
Midfielder Harrison had a good, cool finish ruled out for offside on 70 minutes, after some great build up play involving most of the Rovers midfield and attack.
This was Harrison's last action of the game, as he was replaced in midfield by Steve Jennings and midfielder John Mullin replaced Steve Davies in attack.
Mullin had another weak shot easily saved by Edge, whilst Chris Shuker decided to try and control the ball from a Greenacre cross, when had he vollied, he would only have had to get his shot on target to score.
On 87-minutes, Rovers seeled the win, thanks to Steve Jennings after some great work by John Mullin.
The substitute collected the ball in the Tranmere half, and after chipping the ball over the Bury defender, carried his run on towards the penalty area. The ex-Burnley man took a shot which beat 'keeper Edge, although it was probably going wide, until Steve Jennings slid in and netted from only a couple of inches out to get his first of the season.
Mullin saw another shot saved by Edge whilst Shuker also had a shot saved, and John Achterberg was equal to Tom Youngs' effort in the final minutes.
Overall, Ronnie Moore must be very pleased with the win, in a game that his side dominated. The draw for the next round is at 11am on Wednesday 18th October, with full details appearing on the site as soon as possible.
Jones' Man Of The Match: Although everybody had a reasonably comfortable game, for me, Chris Shuker was man of the match. The scouser got his first goal with a brilliant free-kick and was a thorn in the side of the Bury defence all night.