Salford lost Tim Lafai to the sin-bin in just the third minute for a high tackle on Miller and Leeds quickly took advantage.
Within 45 seconds, Frawley’s dummy gave him space to race in, and moments later Paul Momirovski’s powerful break created a two-on-one on Salford’s left edge, giving Edgell a clear run to the line.
As Lafai returned, it was Salford’s turn for a one-man advantage as Martin, having booted Leeds into a 12-0 lead, was sent to the bin for a dangerous tackle.
The hosts duly cashed in as Watkins burst through from close range against his former side, but Croft raced to the right corner from 30 metres to put the 12-man Rhinos two tries clear.
Salford came again before the break and cut the gap once more when Watkins burst onto a scoring pass at pace to power over.
Video referee Jack Smith chalked off Nene Macdonald’s score to deny Salford a half-time lead, but the hosts upped the intensity after the break and Leeds struggled to live with the pace.
They finally overcame determined Rhinos defence when Jayden Nikorima broke through two challenges and teed up Mellor to finish, with Sneyd’s 56th-minute conversion putting Salford ahead for the first time.
The half-back booted over a penalty before James Bentley became the third man to be sent to the bin, pinged for a high tackle on Joe Shorrocks.
Sneyd’s long-range penalty further turned the screw, but the half-back and Shorrocks both missed drop-goals to take Salford two scores clear.
The 12-man Rhinos had looked drained, but somehow responded late on, with Croft going close after a thrilling break, but Salford held out for a huge win.