An unchanged Stags wasted the best chance of the night when 13-goal Mickey Boulding saw his first half penalty saved by Donovan Ricketts.
Boulding's brother Rory also saw a second half effort chalked off for offside.
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But Bradford also had their chances and Omar Daley rattled a post in their best moment.
In the end defences were on top with Stags extending their Field Mill clean sheet to 368 minutes and City celebrating their first away clean sheet of the season.
But it was a result that left Stags still bottom of the pile.
They had slumped back to the foot of the Football League the night before when both Wrexham and Lincoln won.
On a cold and windy night both sides went at it in an exciting first half which somehow didn't produce a goal.
The second half was a much tighter affair though Stags did dominate for long spells without being able to find an end product.
Former Stags striker Barry Conlon almost gave the Bantams a dream start in the first 30 seconds.
Mansfield were caught pushing forward by a lightening break which saw Daley race down the right, cross long to the far side of the box and Conlon met it with a superb on-target volley that in-form Carl Muggleton had to turn aside for a corner.
On five minutes Mickey Boulding put over a cross from the left by-line that only just cleared Stephen Dawson's head with the goal at his mercy.
Conlon came desperately close again on 10 minutes as a low Guylain Ndumbu-Nsungu ball in from the left went through everyone and 12 yards from goal the big striker curled a finish narrowly over the far angle.
However, Stags wasted the opportunity of the night from the penalty spot on 20 minutes.
Lee Bell's pass sent Rory Boulding into the box where he went down under the challenge of David Wetherall.
After a second's hesitation, the referee pointed to the spot to the visitors' fury.
Mickey Boulding tried to slot the ball home low to the keeper's left but didn't strike it that hard and Ricketts guessed correctly and made a comfortable block.
Within two minutes Kyle Nix took a spectacular tumble in the box at the other end, hoping referee Jarnail Singh might want to even things up.
But while he was unsuccessfully screaming for a penalty and the defence dithered, the ball ran on to Daley who smashed a rocket shot against the near post.
The end to end action continued as Mickey Boulding got into a good position at the other end but scuffed his finish.
Then brother Rory's deflected shot flew up and back down into the arms of Ricketts who juggled nervously with it for a second before holding on tight - one of several moments of worry the eccentric keeper gave his side.
Mansfield won their first corner on 28 minutes which was cleared and Bradford again launched another swift counter which saw them three on two at the other end only for Muggleton's legs to deny Ndumbu-Nsungu's low finish.
On 39 minutes Bell's' through ball put Mickey Boulding into the box and he rounded the keeper but could not find a finish as he went down under the keeper's challenge.
Stags continued to look the team most likely to break through.
No one could get on the end of Gareth Jelleyman's low cross in front of goal. Then Darren Williams was harshly booked for bringing down Mickey Boulding and John Mullins headed well over from Matt Hamshaw's free kick.
Again Dawson was seconds too to get onto the end of Rory Boulding cross right in front of goal as an exciting and enterprising half ended scoreless.
Stags were dealt a blow on 53 minutes when Bell limped off to be replaced by Simon Brown who went wide left with Jelleyman dropping into an unfamiliar central midfield role.
Dan Martin was booked a minute later for a blatant body check on nippy winger Daley as he tried to skip past him again which saw several players race in to square up.
Stags were struggling to make an attacking impact, despite their possession, and Micky Boulding's long range hopeful low effort was wide and carried no real threat.
Hamshaw then sent Mickey Boulding to the right of the box where his cross was fumbled by Ricketts and the keeper held at the second attempt with Dawson on hand to punish any further juggling.
Rory Boulding finally had the ball in the Bradford net from brother Mickey's cross on 65 minutes but an offside flag was already raised.
Matthew Clarke then stormed into the Stags box on a great run and his low cross was hooked clear by Jake Buxton. Then Muggleton had to clutch a backheader off one of his own defenders as the home side survived some nervous moments.
Mullins slipped a neat ball in to Brown just inside the box on 81 minutes but he pulled his shot wide.
Bantams sub Peter Thorne was well over from 18 yards for Bradford as the sides began to run out of steam and ideas.
The visitors did step up a gear near the end and forced two corners while Daley saw a vicious 20 yard blast blocked by a well placed Muggleton as Stags at least held onto their point.
STAGS: Muggleton, Mullins, Jelleyman, Bell (Brown 53), Hamshaw, Dawson, M. Boulding, Buxton, John-Baptiste, Martin, R. Boulding. Subs not used: White, McIntosh, McAliskey, Holmes.
BRADFORD: Ricketts, Williams, Heckingbottom, Wetherall, Daley, Conlon (Thorne 68), Clarke, Phelan (Evans 85), Nix, Ndumbu-Nsungu , Law. Subs not used: Ainge, Penford, Waite.
REFEREE: Mr Jarnail Singh of Middlesex.
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