United went to the County Ground at the weekend in a rich vein of form with a run of three consecutive victories under their belts.
There was one change to the starters from the home victory over Grimsby last week with Joel Senior in for Jordan Gibson.
Ryan Edmondson made a welcome return to the bench, at the expense of Jayden Harris, with the remainder otherwise unchanged.
A run from Hepburn-Murphy earned Swindon an early corner as Mellish stepped in to get in the way. Holy punched, and McCalmont set off on a break. He was blocked as he entered the opposition half.
A quick throw from Senior saw him exchange passes with McCalmont, before Patrick sent it wide. Senior's delivery was too close to Brynn and he dropped low to pick it up.
Hepburn-Murphy was on tbe move again as it flowed from end to end, and he beat two before he drilled it across the face of goal. There was nobody there to finish it off.
There was suddenly space for Charlie Austin to let one go as things tightened up, but he was quickly closed down and Feeney completed the clearance.
Senior was chopped to the floor as he wrestled possession away from Wakeling, but Armer curled it into the keeper's hands from tight by the touchline.
Garner was sharp with another throw from the right flank that put Patrick into the box. He megged the first challenger but Brynn was out to fall on top of it.
Good football from United, with a triangle of passes in midfield, resulted in a corner as Senior's cross was diverted behind. It was floated deep by Armer, but Feeney couldn't quite make the connection he needed and it plopped wide.
Patrick then skipped past his man as he latched onto a ball over the top, but the Robins managed to get it away after McCalmont had try to squeeze a volley through. Senior had half a shout for a penalty when possession was recycled, but the referee showed no interest.
Brynn needed two attempts to grasp a clipped header from Garner as he got on the end of a Moxon free kick, and the forward showed fleet of foot before the break to turn away from two before once again being felled.
Carlisle thought they'd opened things up when Moxon looped a deep free kick towards Huntington inside the area. It eventually came to Garner, with his back to goal, and he cushioned it for Mellish to strike. It wouldn't quite settle and Armer was frustrated when his snapshot was blocked by Hutton.
There was a scare at the other end when Williams and Austin combined to put Mellish under pressure, and his sliced clearance had Hepburn-Murphy on the stretch to head. He couldn't reach it and the moment was gone.
It remained extremely tight at the start of the second period, with Hepburn-Murphy flagged for offside as he tried to bring a spark.
But United took the lead on 51 minutes in some style after Austin had diverted a wicked Moxon corner onto the angle of post and bar. It was clumsily cleared, but Jack Armer was lurking, and he registered an absolute beauty, into the top corner with his right foot, from the edge of the area.
Half of the Swindon crowd thought Austin had equalised just two minutes later when he got on the end of a fantastic free kick, just 12-yards out. His header had power but it zinged the wrong side of the upright.
A patient build-up saw Patrick, Guy and Garner switch the play from left to right, where Senior stepped over to create room for a cross. He ripped it across the face of the target and Armer, who had followed the move, was a whisker away from getting a toe to it.
There was a real chance for the Robins on 62 minutes when a low corner somehow found a way through. Garner stuck a leg out, Wakeling went to ground, and McEachran brought the best out of Holy with a thundering 19-yard bullet of a free kick. The keeper was at full stretch to palm it away from his top corner.
Clayton tugged Feeney to the floor on the 18-yard line on 70 minutes, as the pair chased after a flighted delivery, but Moxon couldn't find a way through the solid wall when he took charge of the set piece.
The Robins threw men forward, but Paul Huntington led by example as he made sure he was first to every ball into the box.
Wakeling sent one wide from range as the red shirts struggled to find a clear sight, but the equaliser came on 80 minutes when substitute Tyrese Shade shielded a deflected effort from Lavinier that came his way. He poked it through the defending legs and into the bottom corner as Holy tried to close the space.
Holy was comfortable with a skewed cross from Darcy in time added on, but there was a late, late chance for the Blues when Brynn miskicked a clearance. Edmondson and Moxon won a corner, and the game was won in dramatic style as Moxon sent it long and deep. With 96 minutes on the clock Ryan Edmondson marked his return with a perfect header which claimed all three points for the visitors.
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Goals
Swindon Town - Shade (80)
Carlisle United - Armer (51), Edmondson (90+6)
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Bookings
Swindon Town - Wakeling (17), Austin (57), Clayton (70), Shade (87)
Carlisle United - Guy (9), McCalmont (56), Moxon (76)
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Teams
Swindon Town - Brynn, Clayton, McEachran, Wiliams, Austin (Lavinier 66), Brewitt, Cain (Darcy 66), Hepburn-Murphy, Wakeling (Jephcott 81), Minturn (Shade 66). Subs - Copland (gk), Adeloye, Jephcott, Kadji.
Carlisle United - Holy, Feeney (c), Huntington, Senior, Mellish, Moxon, Guy, McCalmont (Gibson 68), Armer, Patrick (Gordon 81), Garner (Edmondson 68). Subs - Kelly, Dennis, Whelan, Robinson.
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Match officials
Referee - Sam Purkiss
Assistant Referee 1 - Daniel Flynn
Assistant Referee 2 - Ryan Whitaker
Fourth Official - Abigail Byrne
Observer - Paul Rejer
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Attendance - 9,330 (558)