Leigh have only won one game all season and look doomed to relegation, but they are not going down without a fight.
New boss Steve Bleasdale had them organised and with defender Chris Clarke's arrival on loan from Halifax, they looked more solid at the back and limited the Grecians to only a couple of scoring opportunities.
Fortunately for the visitors they only need one chance, in the 32nd minute, when a couple of defenders hesitated long enough for Danny Clay to push the ball into the path of Sean Devine, whose miscued shot had enough pace to roll past a stranded goalkeeper.
Devine looked set to add a second late in the game but new keeper Jon Worsnop, drafted in from Droylsden to cover for the injured James Mann, produced a superb save.
Exeter were unlucky not to be awarded a penalty when Leigh pushed forward late in the game, trying to grab an equaliser.
They were left exposed at the back and Clarke looked to have brought down Steve Flack in the area, but the referee waved away the players' penalty appeals.
For all Leigh's ability at the back, they were still lacking upfront where they rarely offered a well-organised defensive line any problems.
RMI were limited to long-range efforts and Warren Peyton and Andy Roscoe fired way off target when they had a chance, while Lee Shillito's 25-yard drive gave the keeper little cause for concern.