Swindon avenged their opening day defeat at Priesfield with a dominant second half display to secure the points at the County Ground.
Mark Stimson kept faith with the formation and most of the side that gained a draw at Exeter in their last league match, Josh Gowling the only change replacing Matt Fry at the heart of defence.
Good link up play between Nutter and Jackson produced the games first opening, the latter dragging a low effort wide of Lucas' left hand post on four minutes.
The home side looked confident and thought they had opened the scoring in the 18th minute when Austin tucked the ball under the advancing Julian but his strike was ruled out for a foul on Bentley in the build-up.
In a first half short on decent chances, Charlie Austin went close again on 26 minutes with a speculative effort from 25 yards that was comfortably held by Julian.
On the half hour Gillingham struck the woodwork when a Nutter free-kick over shot the box, falling to Jackson on the edge of the area, his stabbed shot hit Lucas' left hand upright, glancing to safety behind the goal.
Swindon finally took the lead in the 37th minute, Simon Ferry's clever through ball allowing Austin to hold-off Gowling before plating his finish into the far corner.
Four minutes before the break the visitors could have been on level terms as Maher drilled Nutter's corner goalwards but the offside flag was raised against Simeon Jackson, then, at the other end, Alan Sheehan was
unlucky with a curling effort from the left angle of the penalty area that narrowly missed the far post.
Paynter dragged wide from 20 yards, then on the stroke of half time Nutter's free-kick was headed across goal by Gowling, Lewis poked it towards goal and watched in horror as it rolled inches wide of the Lucas right hand post.
Gillingham went on the offensive from the restart with Palmer shaving the upright with a low shot to the near post after a neat lay off by Rashid Yussuff.
Moments later Swindon extended their lead, Scott Cuthbert heading home from McGovern's corner as Maher, Fuller and Julian tried in vain to keep it out.
With the home side growing in confidence Mark Stimson sent on Febian Brandy and switched to a 4-4-2 formation to try and salvage something from the match, he was followed on the hour by Curtis Weston as the manager sought a spark of creativity.
Soon after a neat move involving Nutter and Brandy gave Yussuff a sight of goal but he blazed over from the edge of the box. On 67 minutes Danny Ward collected a weak pass and rifled wide of Julian's right hand post with a low drive from 25 yards.
Further chances went begging for Ward, while John Nutter spared Mark Bentley's blushes, clearing his goalbound header into the stands to prevent an o.g.
On 73 minutes Bentley's foul on sub Pericard gave Billy Paynter the chance to slot home from the penalty spot for the Robins third.
By now Swindon were rampant and Alan Julian pulled off important saves to deny Douglas and McGovern as the home side sought their fourth.
Gillingham grabbed a neat consolation in the 88th minute as Nutter's short corner came to Chris Palmer on the left edge of the box and the full back blasted into the far corner to claim his first senior goal for the club.
Gillingham: Alan Julian, Barry Fuller, Josh Gowling, Mark Bentley (Tom Wynter 76), Chris Palmer, Dennis Oli, Stuart Lewis (Curtis Weston 60), Kevin Maher (Febian Brandy 56), Rashid Yussuff, John Nutter, Simeon Jackson.
Subs not used: Simon Locke, Jack Payne, Luke Rooney, Mark McCammon.
Swindon Town: David Lucas, Scott Cuthbert, Alan Sheehan (Lecsinel Jean-Francois 60), Jonathan Douglas, Gordon Greer, Jon-Paul McGovern, Kevin Amankwaah, Billy Paynter (Ben Hutchinson 80), Danny Ward, Simon
Ferry, Charlie Austin (Vincent Pericard 68).
Subs not used: Phil Smith, Michael Timlin, Sean Morrison, Lloyd Macklin.