The two scored either side of Danny Mayor's strike in a contest which was vital to both sides' chances of avoiding relegation from the Football League.
The Cobblers started well with John Marquis twice testing Shakers goalkeeper Brian Jensen after returning on loan from Millwall for a second spell.
At the other end, Daniel Nardiello was denied by Matt Duke before Bury broke the deadlock in the 18th minute. Hope, who had a spell at Sixfields earlier in the season, was picked out by Mayor and he beat Duke from close range.
Marquis went close to an equaliser when he hit the post and then saw his header saved by Jensen before Bury doubled their advantage 10 minutes before the break. Mayor fired past Duke with a drive from the edge of the box which flew into the roof of the net.
After finding themselves 2-0 down at the break, Northampton failed to offer a serious threat after the restart and it was the visitors who looked the more likely to score. Hope drilled wide from a good position and substitute Rose was denied by Duke.
Ian Morris went close with a header late on as the hosts finally carved out a decent opening, while substitute Gary Deegan forced Jensen into action and Emile Sinclair had an effort blocked.
But Chris Wilder's men were unable to get back into the contest as they saw another chance to get out of the relegation zone come and go. And a stoppage-time goal from Rose wrapped it up for the visitors as he slotted home from close range.