Harry Cornick grabbed the all-important goal just before half-time as he converted from James Collins' low cross, with Bury unable to find a reply in the second 45 minutes.
The Shakers from the first minute were slick in possession and created their first decent opportunity just 30 seconds in as Maynard drove an effort narrowly over Shea's bar as he managed to dart in from the left.
Glen Rea had a great opportunity to open the scoring seven minutes in as he got across his marker well and headed narrowly wide from James Justin's out-swinging corner, sending it wide to the left of Joe Murphy's goal.
Five minutes later the hosts saw James Shea deny them the lead when Byron Moore pulled the ball back from the by-line on the right, Maynard was there to side foot it low from close range, but Shea got down quickly to palm it away, before Bradley cleared.
Ryan Lowe's side continued to look dangerous and were desperately unlucky as Neil Danns was guilty of missing a gilt-edged opportunity. He was found by Chris Stokes on the edge of the box, he turned Bradley, leaving the defender on the ground, the Bury supporters were ready to celebrate as the former Bolton Wanderers man had the goal at his mercy, but he sent it wide to the left of Shea's goal from ten yards.
Town grew into the game after the half hour mark and Rea's lovely ping from the middle of the park found Stacey out on the right, he bolted past Stokes and got a shot away from a tight angle 15 yards out, but Murphy was able to palm it down and gather.
The deadlock was broken with three minutes of the first half to go when Lee played a delightful ball down the left for Collins, he found Cornick at the back stick and the ex-Bournemouth man made no mistake from five yards – poking in for his fifth goal of the season.
He should have doubled the lead and his tally for the afternoon in the first minute of injury time as the Hatters countered from a Bury corner, he played a one-two with Justin as they sprinted forward, but he shanked his effort from 15 yards well over.
The second half lacked the same flow as the first as Bury's energy waned slightly and the Town showed their League One quality.
Telford somehow spurned a great chance for Bury 73 minutes in as Maynard crossed in from the right, the substitute turned well, but got his boot well under the ball and hoofed it into the stand behind the Town's goal.
Collins missed an equally good chance just a minute later as Lee picked out Stacey on the right, he got in behind the Bury backline before crossing it low, Collins timed his run perfectly and did well to get a first-time effort away from all of seven yards but Murphy threw himself across to get his body in the way of the strike as the chance went begging.
The hosts poured forward late on in desperate search of an equaliser, but Nathan Jones' side were able to weather the storm.
The Town will be ball number 46 in the third-round draw tomorrow evening.
Bury: Murphy, Stokes (Miller 83), Thompson, O'Connell, Adams, Mayor, Moore (Telford 58), McFadzean, O'Shea, Danns © (Lavery 74), Maynard.
Subs not used: Moloney, Aimson, Dawson, Dagnall.
Town: Shea, Stacey, Pearson, Bradley, Justin, Rea ©, Mpanzu, Shinnie, Lee (Potts 83), Collins, Cornick (Hylton 87).
Subs not used: Stech, LuaLua, Jarvis, Jones, Sheehan.
Goal: Cornick 42
Yellows: Shinnie, Rea, Mpanzu, Bradley
Attendance: 2,977
Away: 549
