This nerve-jangling victory has left Chris Casper's charges six points above the dreaded drop zone with just two games remaining in the season.
"We took it right to the wire, but the attitude from the players was fantastic," said Casper.
"We came from behind twice and won the game, so that has to be applauded.
"We're not safe, but that has helped. We are going out to win our next two games and to put our relegation battle to bed."
Two goals in the last 11 minutes of the game, the last of which came in the sixth-minute of injury-time, ultimately proved crucial.
Having twice gone behind, goals from David Flitcroft, Brian Barry-Murphy and saviour striker Tipton made sure Casper's men triumphed.
And with fellow strugglers Torquay still below them despite a 1-0 win against Wrexham and Rushden and Diamonds beaten 3-1 at home to Wycombe, the Shakers are now favoured to stay up.
The match started badly for Bury when Chris Brass was unfortunate to deflect the ball into his own net. Neil Wainwright's shot coming off the defender and past goalkeeper Kasper Schmeichel who had no chance.
But then Flitcroft levelled matters with a sweet strike midway through the first half and then Lewis Gobern went close to making it 2-1 in the visitors' favour.
Target
Four minutes before half-time Barry-Murphy saw a shot saved by Darlington keeper Sam Russell, before at the other end Schmeichel did well to deny Jemal Johnson.
Marc Pugh then shot over on the stroke of half-time as Bury looked for the lead.
Pugh had another good chance in the 54th minute, but fired wide until Gobern also missed the target in the 57th minute.
Gobern fired over the bar in the 62nd minute, until Akpo Sodje headed the home side into the lead with 10 minutes remaining.
However, Barry-Murphy found time and space to find the back of the net, before hitman Tipton proved the Shakers' saviour and hero with a winner.
Tipton said: "I've come on and hopefully that's the winner that keeps us in the league.
"We've come from behind twice. It was gut-wrenching to go 2-1 behind, but we've come back and done brilliantly in the game." Tweet

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Great win that should see you safe. Now a good result for SCFC at Torquay and we can all sing, "We'll meet again..."
Cheers,
Phil B^)
PS we still have a saddo posting poison on the Stockport County page calling himself Matt amongst other aliases and pretending to be a Bury fan so the County slag off Bury in retaliation. Happily most, if not all, of us see right through him/her/it and wish you well.......until the next SCFC v Bury game anyway!!
Quiet on these pages isn't it?