Cody McDonald scored the only goal for the Gills, heading home from close range five minutes into the second half at the Proact Stadium. The win extended the Gills' unbeaten run to four games.
Second-placed Port Vale beat Torquay courtesy of a Calvin Andrew strike.
Burton brushed aside Exeter 4-2 to record a seventh successive home league win and move into the automatic promotion places in third.
Michael Symes, Marcus Holness, Jacques Maghoma and Lee Bell struck for Albion before Guillem Bauza and Scot Bennett restored some respectability for the Grecians.
Northampton moved up two places to fourth with a 1-0 win over Plymouth, who dropped to the bottom. Clive Platt scored the Cobblers' winner after 72 minutes.
AFC Wimbledon took advantage of the Pilgrims' loss, and the failures of Accrington Stanley and Barnet to win, to climb off the bottom and out of the relegation places.
Kevin Sainte-Luce scored the only goal 10 minutes from time as Wimbledon won 1-0 at Dagenham and Redbridge.
Accrington, however, still claimed a creditable 2-2 draw against play-off contenders Cheltenham at home.
Cheltenham twice led through Marlon Pack, the first from the penalty spot, but former England international James Beattie responded to both goals for Stanley.
Barnet played out a goalless draw with Rochdale while another side in trouble, Aldershot, won 2-0 at home to a Fleetwood side who ended the game with nine men.
Danny Hylton struck twice for Aldershot before the dismissals of Jamie McGuire and Alan Goodall. Craig Reid also missed a penalty for the hosts.
Goals from Barry Corr and Britt Assombalonga saw Southend come from behind to win 2-1 at Wycombe, who had led early through Dean Morgan.
Oxford's clash with York ended 0-0 while Morecambe and Bristol Rovers drew 1-1. Rovers' Eliot Richards cancelled out a second-minute goal from Jack Redshaw.
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