The hosts raced into a 22-8 lead at the break thanks to tries from Weller Hauraki, Craig Hall, George Lawler and Jimmy Keinhorst, with the Red Devils briefly leading 8-6 after 26 minutes through Rob Lui and Josh Jones scores.
However, Ian Watson's side added 16 unanswered points in the second period as Derrell Olpherts, Ken Sio and Joey Lussick crossed the whitewash as Sio's second conversion with nine minutes left proved decisive in the 24-22 victory.
Salford returned to winning ways after last week's 46-14 hammering at home to Leeds to secure their third triumph from four matches, while the Robins have opened their season with two victories and two defeats.
Hull KR moved in front after seven minutes when former Salford forward Hauraki got on the end of Josh Drinkwater's grubber kick to crash over, with Ryan Shaw adding the extras.
Salford hit back when Lui collected a Jackson Hastings pass before throwing a dummy to power his way before Jones scored his third try in as many games to put the visitors into an 8-6 lead.
Three tries in the eight minutes at the end of the first half put Rovers in control as Drinkwater's perfectly-weighted kick found Hall to dot down in the corner before Lawler stretched over.
Keinhorst then scored a breakaway try in the 37th minute to extend Hull KR's advantage to 22-8, but that was as good as it got for the hosts.
Salford went in search of closing the gap and they did just that when great interplay between Jackson and Lui sent Olpherts over to reduce the gap to 22-14.
Sio finished off a fine Salford move when quick hands from Jackson, Lui, Niall evalds and Junior Sa'u put the winger in the corner with 16 minutes left, but he could not convert his own score which would have brought the away side level.
However, Sio held his nerve seven minutes later when he kicked the match-winning conversion after hooker Lussick bundled his way over from close range to seal a great comeback victory.
