Rose drove a low 70th-minute free-kick beyond Swindon goalkeeper Wes Foderingham to hand new manager Joe Dunne his second victory from as many games in charge.
Swindon - who had only lost once at the County Ground in 14 months - paid the price for a toothless attacking showing in a game of few clear-cut chances.
Robins striker Andy Williams saw his early shot on the turn smothered by Colchester goalkeeper Mark Cousins while Anthony Wordsworth sent a shot just over the home crossbar.
Jamie Collins, Williams and Matt Ritchie all failed to hit the target for Di Canio's side with presentable second-half openings.
Substitute Freddie Sears almost doubled Colchester's lead late on after seizing on a defensive error but could only drag his shot wide.