Hull KR led 34-14 until the Reds rallied to score four converted tries without reply. Defeat was tough on Michael Dobson who piled up 18 points with two tries and five goals, but the Reds had the man of the match in Mark Sneyd.
Both sides finished with 12 men after Reds prop Lee Jewitt and Rovers forward Adam Walker were sent off after a punch-up.
Salford, watched for the first time by new Director of Rugby Graham Lowe, fell behind after seven minutes when Dobson and Greg Eden combined to send David Hodgson over in the corner against his old club. Dobson booted the touchline conversion and Rovers would have gone further ahead had Omari Caro not spilled Liam Salter's pass with the line open.
The Reds joy was shortlived after an Ashley Gibson error led to Dobson touching down for a converted try to make it 12-0. Salford hit back with Sneyd's long pass allowing Gibson to cut inside for his third try of the season. Sneyd was involved again with Gibson beating Caro to a neat kick for his second touchdown in four minutes.
But Sneyd was again off target with the conversion attempt to leave the Reds trailing 12-8. Debutant Darrell Griffin was introduced after 25 minutes and made his presence felt with an early barnstorming run. But it took an audacious move by Sneyd which fired Salford ahead for the first time. The scrum half stole the ball off Eden to dart over and boot the conversion to make it 14-12.
Eden made amends when he took Cory Paterson's smart pass to send Dobson streaking over within a minute to restore the Rovers lead at 18-14. The final delivery looked suspiciously forward but there was no dispute about Eden's pass to Caro which stretched the advantage to 24-14 three minutes before the break. Dale Ferguson had a try disallowed for a foul on Jack Murphy from Dobson's high kick.
Eden took only four minutes of the second half to put Rovers into a more commanding position when he took Dobson's pass to round Jack Murphy and make it 30-14. Eden bagged a second before Andrew Dixon notched his first since his move from St Helens.
Battling Salford cut the gap to 34-26 through Murphy's converted try on the hour. Williams went close for Salford with Rovers under the cosh and it was no surprise when Wayne Godwin burrowed over.
Sneyd's conversion made it 34-32 with eight minutes left before Salford regained the lead when Ryan McGoldrick's kick bounced nicely for Williams to score the decisive try. Even after all that, there was still time for more drama, with Jewitt and Walker seeing red for a fight at a scrum.