RMI produced a backs to the wall performance to hold out a determined Forest Green side who dominated most of the game but squandered a series of chances.
Rovers, with only one defeat in 15 games, posed all the questions until the 74th minute when Martyn Lancaster's powerful header from Andy Heald's curling free-kick proved to be the winner.
Forest Green started the livelier of the two sides but the bumpy Hilton Park pitch came to Leigh's rescue when Darren Jones had an early chance, but the ball bobbled up and his shot flew over the target.
Kayode Odejayi caused panic in the Leigh ranks when he won a header and raced into the clear but dallied too long over his shot which allowed Lancaster to recover from his earlier error.
Forest applied almost continuous pressure but Neil Grayson fired straight at the keeper, Martin Foster fired just wide and a ferocious 20-yard volley from Alex Meechan was tipped over the bar by Stuart Coburn.
Alex Sykes saw a free-kick skim the bar before Coburn rescued Leigh again when he tipped a Foster free-kick around for a corner.
Chances were at a premium for RMI, but when it did come their way they did not trouble the keeper and Lancaster was guilty of blasting the ball over the target when a rare chance came his way.
Leigh were hoping for better fortunes with the wind at their backs after the break, but it was Rovers who continued to create the better chances until Lancaster's shock goal.
It was clearly not going to be Forest Green's day when five minutes from time Jones outjumped the defence only for his header to smash against the crossbar and in the final minute Jon Richardson missed an open goal after Foster's free-kick had beaten the Leigh cover.