The champions looked to have taken the derby spoils courtesy of Peacock's first try against his old club which put them into an 18-12 lead but he was yellow-carded for dissent three minutes from the end and the 12-man Rhinos, for whom Kallum Watkins and Mitch Achurch also crossed, could not hang on.
Leeds skipper Kevin Sinfield had what would have been a match-winning drop-goal ruled out while he was wide with another attempt. But it was just reward for Bradford's Adrian Purtell, who scored his first try since suffering a heart attack in the aftermath of his last meeting with Leeds 10 months ago, while Tom Olbison also went over for the Bulls.
The Bulls were good value for their point, especially after recovering from a calamitous start. The Rhinos gained possession from their own kick-off when the ball bounced wickedly into the grateful hands of onrushing second rower Jamie Jones-Buchanan and full-back Watkins took Sinfield's pass to score from the next play.
The champions looked to have added a second try before Bradford could recover, with winger Ryan Hall touching down after full-back Brett Kearney had been deceived by Joel Moon's grubber kick. Video referee Ian Smith disallowed the try because Jones-Buchanan was offside from the kick but it looked a harsh call since he was not directly involved.
The Bulls made the most of the let-off as they gradually worked their way into the game to take a 12-6 lead by the quarter mark. Purtell took Elliott Whitehead's superb offload to finish off a scintillating move which featured six pairs of hands and second rower Olbison caught the Leeds defence napping by scoring from dummy-half.
Jamie Foster added both conversions and the Bulls almost extended their lead on 22 minutes when stand-off Danny Addy forced his way to the line only to lose control of the ball. Leeds drew level when forward Achurch proved unstoppable after taking Sinfield's pass 10 metres out to crash over and Sinfield kicked his second goal.
The game swung back in the Rhinos' favour in the third quarter after the Bulls were starved of possession. Jones-Buchanan got over the line on 51 minutes without being able to ground the ball but there was no stopping Peacock moments later as he took three defenders over with him to register his second try of the season.
Sinfield's third goal made it 18-12 but the score only served to spur on the visitors, who put the Leeds defence under enormous pressure. Sinfield thought he had made the game safe with a drop goal nine minutes from the end but the score was disallowed for Peacock's push and the Leeds skipper was also wide with another attempt.
In a dramatic finish, Jarrod Sammut was brought down inches short of the line and Leeds were reduced to 12 men when Peacock was shown a yellow card for dissent. Bradford immediately took advantage of the extra man, with Sidlow going through a yawning gap in the Leeds defence for his side's third try and Foster's third goal locked up the scores.
Zeb Taia crossed for two second-half tries as the Catalan Dragons moved top of Super League by downing London Broncos 30-24.
Taia's efforts proved decisive in a contest that swung this way and that from first minute to last in front of a sparse crowd at Esher.
Kieran Dixon, Tony Clubb, Michael Witt and Chad Randall all went over for the hosts while Morgan Escare, Steve Menzies, Frederic Vaccari added to Taia's two tries for rhe Dragons.
The men from the capital were rewarded for their strong start on five minutes, Dixon swooping on a loose ball and racing home from inside his own half. But that lead proved short-lived, Escare showing a clean pair of heels to storm home after a clever pass from Leon Pryce.
Midway through the half though the Broncos retook the lead, Clubb powering over from close range on his 150th appearance for the club. Yet once again the Dragons were to hit back within minutes, Menzies crossing in the corner after more good work from Pryce.
In bitterly cold weather both sides were contributing to an entertaining, and open, contest, but with no further scoring the Londoners entered the break 12-10 ahead.
Once again the home side were quickest into gear after the interval, stretching their lead on 42 minutes when Shane Rodney's break allowed Witt to touch down at the posts. But, as in the first half, the Dragons quickly struck back, Vaccari racing in at the corner after a neat handling move.
The tries kept coming, Randall the next to cross for the hosts, racing through a huge gap after his dummy fooled the Dragons defence. But try as they might the Broncos could not pull clear, Taia making it a one-score game yet again when he gathered a Pryce kick to ease over.
As the game progressed the Dragons were gaining the upper hand, the Broncos conceding penalty after penalty, Thomas Bosc electing to kick one to make it 24-24 on 64 minutes.
And with the French finally in the ascendancy, Taia nudged them ahead for the first time when he pounced on a loose pass to race over. Dixon then had a late score chalked off for a forward pass as the action swung to the other end.