Bury had the lions' share of the chances in this action-filled mid-table clash, but some excellent saves from Bristol keeper Ryan Clarke made sure that the visitors snatched a point for their battling efforts.
Bury started much the brighter and the home side should have taken the lead in the fifth minute.
Defender Jon Cartledge headed against the underside of the bar after an excellent corner from David Flitcroft on the left wing.
And Graham's Barrow's team again went close in the ninth when David Nugent broke free and had a shot charged down in front of goal.
Nugent had a second bite of the cherry two minutes, but this time the striker was denied by a superb one-handed save from Bristol goalkeeper Clarke.
Tom Kennedy had a free-kick saved as Bury continued their dominance and Flitcroft saw his powerful right-footed shot deflected for a corner.
Debutante Simon Whaley nearly gave Bury the lead they so desired three minutes before half-time.
The midfielder found time and space in the box, but when Nugent squared the ball to him after a clever run from Chris Porter, he scooped his left-footed shot over the bar.
Flitcroft had another good chance in first-half injury time, but directed his header agonisingly wide from Kennedy's corner from the right.
The resumption saw Porter have a left-footed shot saved and Nugent fired wide of the left post.
Cartledge then hit the bar for the second time in the match with a close range effort, but still Bury could not hit the back of the net.
Skipper Matty Barrass shot wide shortly after the hour mark and it was then Bristol had their first chance of the game.
Striker Junior Agogo toiled under pressure from defenders, but found space to unleash a left-footed shot from the edge of the six-yard box.
But Agogo's shot hit the underside of the bar and rebounded to safety.
At the other end Nugent shot wide of the target after a good flick on from Whaley, while Cartledge had another great chance, only to see his header saved at the near post by Clarke.
Agogo had Bristol's second chance of the match in the 70th minute, but his shot-come-cross was well-saved by Bury keeper Glyn Garner who got down well to deny the centre-forward.
Garner was there again three minutes from time to spare Bury's blushes with a superb save from Agogo, who broke free at the back post to fire on target.
Bury boss Barrow said: "It was one of those games. I thought we'd missed the boat after the amount of chances we missed in the first half.
"Glyn (Garner) had to make a couple of good saves for us at the end and Bristol could have nicked it.
"But we were always the better team and it's pleasing we are making progress. It's the most chances we have created in a while.
"We competed well against a team who have spent a lot of money on the field - and we did it well." Bristol co-manager Russell Osman was just happy that his team had managed to take a point having started so badly to the game.
"It's been hard to motivate the players, but we've done enough to stay up, so we're happy," admitted Osman.
"We just needed the players to give a bit more in the second half. We were second to everything in the first half and it took our players 45 minutes to get going.
"I think the players were cruising a little bit, but we did get a response from them in the second half."