Karl Fitzpatrick broke away for the opening score on 13 minutes and was soon followed by David Hodgson.
Henry Fa'afili gave Warrington some hope but two quick tries at the end of the first half from Luke Robinson and Stuart Littler put Salford in control.
The home side tried to push forward in the second half but the Reds defence held firm for the points.
There were injury concerns for Warrington, who lost influential half-back Lee Briers before kick-off with a dead leg while Chris Bridge was carried off with an ankle injury during the opening half.
Salford coach Karl Harrison "To get two points away from home is a bonus and I thought we were well worth it.
"With the ball I thought we were quite pathetic at times but defensively to keep a team like Warrington to one try is a great achievement.
"We are a very fit, very healthy and very hungry team and we do a lot of defensive work - and we needed to because we kept giving the ball to the opposition."
Warrington: Grose, Fa'afili, M. Gleeson, Kohe-Love, Reardon, Bridge, Sullivan, Leikvoll, Clarke, Wood, Swann, Wainwright, Westwood.
Replacements: Noone, Rauhihi, M. Gleeson, Parker.
Salford: Fitzpatrick, Hodgson, Littler, Moule, Wilshere, Dunemann, Robinson, Coley, Alker, Highton, Brocklehurst, Sibbit, Finnigan.
Replacements: McGuinness, Clayton, Charles, Haggerty.
Referee: R Laughton (Barnsley).
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