Bradford (10) 26
Tries: Menzies, Finigan 2, Reardon, Whitehead Goals: Addy 3
Salford (18) 30
Tries: Broughton 2, Tyrer, Gibson, Fitzpatrick Goals: Holdsworth 5
Broughton has scored 10 tries in 18 Super League games for Salford
Jodie Broughton ran in two tries as Salford inflicted a ninth straight defeat on Bradford, whose late fightback proved to be in vain.
Salford led 30-10 in the second half, but two-try debutant Vinny Finigan, Stuart Reardon and Elliott Whitehead crossed to set up a grandstand finish.
Broughton, Ste Tyrer and Ashley Gibson all crossed for the Reds, before Steve Menzies and Finigan pulled it back.
Broughton put the Reds 20 points up, before their late scare.
Salford, who started the day second from bottom, got off to an ideal start with Broughton travelling over 100 metres and taking advantage of some poor Bulls defending.
Daniel Holdsworth converted and two minutes later their lead was doubled when a clinical Gibson ball put Tyrer in, with Holdsworth again adding two points.
Bulls started putting the pressure on and Menzies crossed, but Gibson capped off a move started by a Heath L'Estrange error to re-establish Salford's dominance, with Holdsworth again converting.
Finigan marked his Bulls debut with a try, which Danny Addy converted - to cut Salford's half-time lead to 10. But seven minutes after the break Jeremy Smith sent Fitzpatrick through the post, with Holdsworth again proving deadly with the boot.
Broughton then appeared to put the game beyond doubt after running from his own half and beating Finigan for pace, with Holdsworth taking his tally for the game to 10 points.
But Bradford, beginning life after England coach Steve McNamara, fought back and Reardon squeezed over in the corner to cut the deficit.
Then, Whitehead finished a flowing move involving Craig Kopczak, Chris Nero and Andy Lynch and Finigan doubled his tally from Brett Kearney's offload - with Addy scoring both goals.
But Salford held on to move up to 12th, with Bradford now three points off the play-off places.
Bradford caretaker boss Lee St Hilaire:
"I thought we contained them very well until Jodie's intervention.
"We had a tough start to the week, losing our head coach, and we've had a tough end to the week as well.
"Steve Mac has laid a great foundation at Bradford Bulls and the club is going to reap the benefits in the years to come.
"Then we lost Dave Halley, who had a bereavement to cope with, and I think that's a reflection of the game.
"We started it badly, had some good bits in the middle and were just unlucky in the end."
Salford City Reds coach Shaun McRae:
"It should have been clinical at 30-10, I've just given the guys a real pat on the back for efforts in that respect but I can't ignore the last 15-20 minutes.
"We really took our foot off the gas a bit but that's in no way disrespectful to Bradford, they played catch-up, they had momentum and were playing stronger.
"When it got to 30-26 I thought 'here we go, this is a game we don't deserve to lose' but I think overall we were probably the better side and deserved to win the game."
Bradford: Finigan, Sheriffe, Platt, Nero, Reardon, Kearney, Addy, Lynch, L'Estrange, Kopczak, Whitehead, Langley, Menzies.
Replacements: Godwin, Worrincy, Crossley, Hall.
Salford: Fitzpatrick, Broughton, Henry, Gibson, Tyrer, M Smith, Holdsworth, Cashmere, Alker, Neal, Sibbit, Littler, Swain.
Replacements: J Smith, Sidlow, Parker, Adamson.
Referee: I Smith (Oldham)