Leigh scored two tries in the opening six minutes and Rovers never recovered, though they put the visitors under pressure at times in the first half.
The 38-12 defeat left Rovers still looking for their first win under coach John Duffy.
Daniel Mortimer opened the scoring after a one-two with Atelea Vea and then created the second try for Gregg McNally with a grubber kick close to the line.
Josh Drinkwater converted both and Leigh led 12-0 before Rovers - who lost all seven of their fixtures in the middle-eights last year - knew what had hit them.
To their credit the Championship side rallied and cut the gap when Darrell Griffin crashed over from close range and Ian Hardman added the extras on 19 minutes.
Luke Briscoe went close for Rovers before Mitch Brown dived over at the corner from McNally’s long pass to increase Leigh’s lead early in the second quarter.
Rovers - and referee James Child - thought they had pulled a try back six minutes before the break, but video official Phil Bentham spotted a knock-on by Briscoe as he stretched over from John Davies’ long pass.
Frankie Mariano was denied a try at the start of the second half when Andy Bostock’s pass was ruled forward and that was Rovers’ last chance to get back in the game.
Mariano made a fine tackle to keep Mortimer out at the other end, but Leigh went further ahead seven minutes after the restart through Liam Hood who dashed over from acting-half for a try which Drinkwater improved. That made it 22-6 and effectively put the game out of Rovers’ reach.
Leigh scored back-to-back tries through Matty Dawson and Drinkwater - who converted the first of them - just before the hour.
Misi Taulapapa capped a fine performance by scoring a consolation try off an offload from Rovers debutant Connor Farrell, improved by Hardman.
But Richard Moore was sin-binned for a foul on McNally who took revenge by scoring Rovers’ seventh try and Drinkwater’s fifth goal completed the scoring.
There were scuffles between stewards and a small number of Leigh fans after the final whistle.