Ipswich loanee Murray came on for Jake Jervis in the 80th minute and two minutes later, netted the winner to decide a match between League Two’s form sides.
Matt Ritchie’s corner from the right was put back into the box by Lander Gabilondo and Murray headed the ball into the top-right corner for his first goal for Paulo Di Canio’s side.
The Shrimpers had taken the lead in the 62 minute when Sean Clohessy’s diagonal ball from the right was bundled down by Luke Prosser and Blair Sturrock prodded the ball home from six yards.
But Swindon got on terms nine minutes later as skipper Paul Caddis scored with a great curling shot from just outside the box.
The result denied Southend a club-record 18-game unbeaten record and stretched Swindon’s run to 13 matches without defeat. It followed a good few days for the Robins, who will tackle Barnet in the area final, after they knocked Southend’s Essex rivals Colchester out of the FA Cup on Saturday with a 1-0 win.