Di Canio led an army of volunteers to clear the County Ground pitch of snow on Saturday and was rewarded with a comfortable win, which might have been greater but for some important saves by Shrews goalkeeper Chris Weale.
Martin, on loan from Norwich, got on the scoresheet in the 49th minute after a lifeless first half of few opportunities at either end.
His goal came from the penalty spot after Andy Williams had been fouled by Connor Goldson, sending Weale the wrong way with his kick.
Williams got Swindon's second four minutes later after Matt Ritchie's cross had been played into his path by Gary Roberts.
Shrewsbury should have pulled a goal back in the 83rd minute after Alan McCormack had fouled Jon Taylor but Matt Richards' penalty was saved by Wes Foderingham.
Weale saved from Danny Hollands and Ritchie and near the end Darren Ward struck a Shrewsbury post.
Any reproduction, publication or redistribution of this material without the written agreement of 365 Media Group is strictly forbidden.