A scrappy game improved after the interval when Jay O'Shea and Armand Gnanduillet secured the points for the home side.
Rovers came into the game brimming with confidence and they almost went ahead in the ninth minute when Eliot Richards' 25-yard shot was brilliantly pushed onto a post by Tommy Lee.
Chesterfield skipper Sam Hird had a couple of headers which Steve Mildenhall comfortably saved but the a defensive error let the home side in eight minutes into the second half.
Lee Brown's clearance only reached O'Shea on the edge of the area and the midfielder smashed a shot into Mildenhall's top right corner.
Rovers almost conceded again in the 63rd minute when Jack Lester headed wide at the back post but Chesterfield made it two when Gnanduillet evaded two defenders to score with a low shot.
The visitors hopes of a comeback ended when another strike by Richards was turned behind by Lee.