With Swansea and Doncaster, the only two clubs above them in the table, losing, Carlisle moved a step nearer a coveted automatic promotion spot.
In a tremendous match full of incidents and chances, Swindon came back from a goal down and looked to be heading for an impressive win.
Carlisle's real hero was their goalkeeper Keiren Westwood who made a series of outstanding saves, including a triple-block in the 30th minute.
Carlisle took the lead against the run of play in the 54th minute when Grant Smith, a former Swindon player, crossed from the left and Marc Bridge-Wilkinson stole in to score with a close-range header.
But Swindon fought back to level in the 61st minute when a corner from Anthony McNamee was forced over the line by Simon Cox for his sixth of the season.
Both sides had chances before Swindon went ahead in the 80th minute when a long clearance found Blair Sturrock who held off Murphy's challenge before beating Westwood with a right-foot shot from just inside the area.
But just when it looked as if Swindon would hold on to win, resilient Carlisle maintained their promotion push when a corner on the right by Simon Hackney was headed in from eight yards by Murphy.
Westwood did not deserve to be on the losing side after his superb display.
The Carlisle goalkeeper denied Cox who was clear in the 35th minute and then was at full-stretch to keep out a well-placed free-kick from Michael Timlin.
The only time Westwood had any luck was in the 57th minute when Timlin rounded the keeper but could not turn the ball in from a difficult angle.