The Red Devils were trailing 8-6 early in the second half of a scrappy contest at the AJ Bell Stadium when 36-year-old Brown jinked his way through for a superb try.
Centre Inu, 34, then produced a sensational interception to set up a breakaway try for Ken Sio and from there Salford did not look back, powering on to a 34-8 win.
Brown claimed a second try for good measure in a strong finish with further touchdowns coming for Chris Atkin and Harvey Livett. Inu finished with five goals.
It was a first victory in four games for new Salford coach Richard Marshall but a result that left the Centurions still looking for their first points since their elevation to the top flight.
Leigh’s night hardly got off to the best start when Lewis Tierney was injured in the warm-up and had to be replaced by Keanan Brand, but they opened positively and edged a tight first half.
The Centurions handed a debut to former Wigan and Warrington centre Anthony Gelling on the right but they soon found attacking down the left their most productive avenue.
Iain Thornley found a gap to put Matty Russell over in the corner for the opening try and Salford struggled to build any momentum, not helped by a lengthy stoppage for an injury to Matt Costello.
When they did finally find a way through midway through the first half, Brown was involved, passing inside for James Greenwood to skip a challenge and score under the posts.
Salford looked to apply more pressure but Leigh held firm and broke out from a penalty to reclaim the lead before the interval.
Again the damage was done on the left with Thornley this time breaking through to touch down.
Thornley might have had another but spilled the ball and Ben Reynolds then had a drop goal attempt blocked in a frantic final play of the first half.
The second half was a different affair as Brown changed the complexion of the game with a flash of inspiration to score.
Better was to follow for Salford as Inu leapt out to intercept a Reynolds pass. He failed to take it cleanly but was quick-thinking enough to flick to Sio, who raced the length of the field.
That broke Leigh’s spirits. Brown put the result beyond doubt with his second of the night and he was also involved in the attacks that saw Atkin and Livett score as the Centurions tired.