Australian winger Scott Donald struck twice in the second half as Leeds ruined Bradford's party and kept the pressure on champions St Helens for top spot in the engage Super League.
The Bulls were on course to celebrate their centenary in style after dominating and leading for most of a rugged West Yorkshire derby played in front of a season's-best crowd of 18,195 at the Grattan Stadium, Odsal.
But Donald twice pounced on Danny McGuire's kicks to the corner to score crucial tries, the second just 10 minutes from the end, to put his side in front for the first time.
A touchline conversion from Kevin Sinfield extended the visitors' lead to four points but Bradford centre James Evans pounced for an equalising try two minutes from the end to tie the scores.
Bulls captain Paul Deacon had a chance to secure both points but he was wide with the difficult conversion.
Leeds still need a point to guarantee a top-two finish but they will see this as a point won and for that will be indebted to former Bradford captain Jamie Peacock, who literally put his body on the line to save two tries.
The visitors were hardly in the game in the first half as the Bulls dominated from the kick-off.
Outstanding prop Andy Lynch went close on four minutes, losing the ball as he stretched out for the line, and Bradford ought to have made more of a clean break by Deacon.
Second rower David Solomona reached the line 14 minutes in only to be prevented from grounding the ball by Peacock's desperate challenge but the pressure paid off when Lynch powered his way to the line past five shell-shocked defenders.
Deacon, returning to the side after a three-match absence, added the conversion and increased his side's lead to 8-0 25 minutes in after he had been tackled late by Leeds second rower Gareth Ellis.
Leeds were given a great lead by hooker Matt Diskin but their inability to hold on to the ball prevented them from building any momentum until the final five minutes of the half.
The Bulls withstood an assault on their line up to the break but they cracked in controversial circumstances early in the second half.
Rhinos stand-off Danny McGuire offloaded from a tackle just short of the line to create what looked like a straightforward tryscoring chance for the supporting Diskin.
But Diskin, who had been held on his back only two minutes earlier, saw the ball spew from his grasp and video referee Richard Silverwood awarded a penalty try, ruling the ball had been stripped by Bradford full-back Michael Platt in a two-man tackle.
Kevin Sinfield's goal cut the gap to just two points but the home side responded immediately, with centre Ben Harris touching down after right winger Marcus St Hilaire collected Iestyn Harris' floated kick to the corner.
Left winger Nathan McAvoy thought he had opened his tryscoring account for the season five minutes later when he was released by David Solomona but Peacock once more came to Leeds' rescue.
The Rhinos then hit back, with Donald winning the race to McGuire's kick to the line for his first try, but Sinfield squandered two chances to level the scores when he missed the conversion and a subsequent 40-metre penalty.
Bradford had been in control for most of the match but it was turned on its head 10 minutes from the end when Donald went up with St Hilaire in pursuit of McGuire's lofted kick and was awarded the try by the video referee.
Sinfield added the goal for good measure but Bradford had the final say when Solomona's long pass found Evans in space and he crossed for the game-levelling score.