Bradford's hopes of a confidence-boosting win before Saturday's Carnegie Challenge Cup semi-final against St Helens were shattered by bottom club Salford.
The Reds moved a point behind Hull KR after a shock but deserved victory inspired by Luke Robinson.
Salford included new captures Mike Wainwright and Richie Barnett from Warrington alongside Castleford's Tommy Saxton and they all played their part in a battling effort.
The Reds, who had won only one of their three matches since Shaun McRae succeeded axed Karl Harrison, were forced to move for reinforcements with Paul Highton, David Hodgson, Lee Jewitt, Ian Sibbit, Mark Edmondson and Karl Fitzpatrick sidelined.
McRae dropped Luke Dorn to sub but Bradford were lacking only Jamie Langley as they chased their seventh win in eight games.
Salford, who were thrashed 56-18 at Odsal in March, started brightly although David Solomona was held up over Reds line.
Bradford had a let-off with Marcus St Hilaire forced to clear from Robinson's kick before the Bulls had a Tame Tupou try ruled out for Solomona's forward pass.
Michael Korkidas and Andy Coley caused problems for Bradford with their forceful surges before the Reds snatched the lead with video referee Ashley Klein ruling Gray Viande just managed to touch down from another probing Robinson kick.
John Wilshere's touchline conversion bounced in off the post to fire his side into an unexpected 6-0 lead after 21 minutes.
Wilshere slotted over a penalty just before the break to make it 8-0 when Paul Deacon was adjudged to have obstructed the Salford marksman chasing Robinson's kick.
Bradford were in more determined mood from the restart and Deacon's 40-20 allowed Tupou to cross after 51 minutes.
Deacon booted the touchline conversion to cut the deficit to 8-6 before Tupou took Solomona's pass to crash over for his second try and put the Bulls in front for the first time.
Salford refused to concede defeat and went back in front when Aaron Moule slipped round St Hilaire for the supporting Dorn to send Robinson over and Wilshere added the goal.
But Bradford tossed away a late chance to snatch at least a draw when James Evans threw a pass behind Tupou with the line at his mercy.