The Addicks impressed in the first half, hitting the woodwork on two occasions, but could not break the deadlock against a Bantams side who are unbeaten at home this season.
And despite being reduced to 10 men in second-half injury time, Karl Robinson’s first league game in charge provided plenty of positive signs, even with a hugely depleted squad at his disposal.
Charlton’s squad had been hit by illness and injury with Declan Rudd, Chris Solly, Ricky Holmes and Ahmed Kashi remaining sidelined and, on top of that, a sickness bug had swept through the squad, ruling out Ademola Lookman and Morgan Fox, as well as leaving many other first-teamers struggling.
Dillon Phillips retained his place in goal, protected by a back four of Ezri Konsa, Jason Pearce, Bauer and Adam Chicksen.
Fredrik Ulvestad and Andrew Crofts, who again captained the Addicks, occupied their regular engine-room berths, while a quartet of Jordan Botaka and Nicky Ajose were deployed out wide. Josh Magennis and Lee Novak started up top.
On the bench, there was some welcome news as Johnnie Jackson and Kevin Foley both returned from injury, while there were also spaces for academy graduates Brandon Hanlan and Josh Umerah.
With the match underway, Charlton suffered yet another blow early on with Pearce suffering an injury after just two minutes, forcing his withdrawal and allowing Jorge Teixeira to emerge from the bench for his first league appearance of the campaign.
The hosts made a bright start, enjoying much of the early possession with James Hanson forcing Phillips into a good, low save with six minutes on the clock.
Three minutes later, at the other end, the visitors’ first foray forwards could well have resulted in a penalty when Magennis appeared to have been hauled down in the box, but although referee Mr Haywood had a good view of the incident, he opted to wave play on, much to the surprise of the travelling fans sat right behind it.
The Addicks had sprung into life by the 13-minute mark with Magennis causing problems again for the Bantams’ defence. The Northern Ireland international powered past his marker and sent a low shot across the dive of the helpless stopper Colin Doyle, only for the ball to bounce back off the post and along the line in agonising fashion.
Moments later, Karl Robinson’s men hit the woodwork again. This time, Botaka worked his magic down the right, jinking into the box before bamboozling Doyle with a looping effort which came crashing back off the crossbar.
The spell of pressure continued and Magennis once again found Doyle backpedalling when he sent Ajose’s cross looping inches over the crossbar with a smart header, before being denied again, this time by Stephen Darby, who was on hand on the line to clear the striker’s goalbound effort after Novak’s hard work had allowed him to pull the ball back in the box.
Indeed, Charlton could well have been three goals up inside 20 minutes had luck been on their side, although Bradford provided a reminder of their own attacking threat on 27 minutes when Josh Cullen fired narrowly wide of the upright from 25 yards.
On 38 minutes, Phillips stood up to another Bradford test when Jordy Hiwula broke into the box. With just the keeper to beat, he was denied brilliantly, keeping the scores goalless going into the break.
Five minutes after the restart, Botaka tried his luck with a superbly-struck drive from 25 yards which looked to be heading into the bottom left-hand corner, only for Doyle to get down well to tip it around the post with a great save.
At the other end, the hosts came close themselves with an effort from similar distance, with Hiwula’s well-hit shot spinning just wide.
The tricky attacker came close again just before the hour mark, beating the offside trap, but not Phillips, who made yet another fine save to keep the Bantams out.
The second half had been a much cagier affair to that of the first, but as the game entered its final stages, the action began to stretch somewhat.
Filipe Morais’ cross-cum-shot nearly caught out Phillips, but the impressive, young stopper did well to claw it away from the top corner and the Addicks would have to have their backs against the walls for the four minutes of additional time after Bauer was given his marching orders for a second bookable offence, stepping in front of a Bradford attacker to prevent him having a clean run on goal.
Charlton, however, stood firm to earn a good point on the road, with the Londoners now unbeaten in five and boasting a record of just one defeat in their last 11 games.
Charlton:Phillips, Konsa, Pearce (Teixeira 3), Bauer, Chicksen, Crofts, Ulvestad, Botaka (Hanlan 90), Magennis, Novak, Ajose (Jackson 76).
Subs (not used):Mitov, Johnson, Foley, Umerah.
Booked: Magennis 24 (foul on Vincelot), Konsa 61 (foul on Meredith), Bauer 69 (foul on Hiwula)
Sent off: Bauer 90+1 (second bookable offence)
Bradford City: Doyle, Darby, Meredith, Law, Vincelot, Marshall (Morais 88), Dieng, Hanson, Hiwula (McNulty 83), Cullen, Knight-Percival.
Subs (not used):Sattelmaier, Vuckic, McArdle, Devine, McMahon.
Booked:Knick-Percival 76 (foul on Magennis)
Referee:M Haywood
Att:17,968 (618 from Charlton)