Matt Ritchie and Andy Williams put last season’s League Two champions two goals ahead inside half an hour.
Ritchie added a third when his free-kick found its way into the net midway through the second-half to seal the win for the Robins.
Substitute Adam Rooney made it four when he converted a low cross in the final ten minutes.
The Cherries’ best chances fell to Matt Tubbs but he couldn’t find the net for the third time in as many games.
Manager Paul Groves made three changes following the midweek draw with Brentford, bringing in Wes Fogden, Steve Cook and Donal McDermott.
Simon Francis, Harry Arter and Marc Pugh missed the trip to the County Ground through injury after picking up injuries on Tuesday evening.
Swindon made a bright start to the game and James Collins shot wide from close range after finding himself found himself unmarked in the penalty area, while Eunan O’Kane sent in an effort from distance.
Di Canio’s team took the lead with just under quarter of an hour gone as Ritchie found the net from 25 yards with a spectacular strike.
The highly-rated winger had a chance to score his and Swindon’s second of the afternoon shortly afterwards, but he dragged his shot wide from the edge of the area.
Captain Miles Addison went close to putting the Cherries back on level terms with his backpost header, but the Robins extended their lead with 25 minutes on the clock.
Striker Andy Williams escaped his marker and guided Gary Roberts’ cross past Shwan Jalal from ten yards out.
The visitor’s best chance of the first-half fell to Tubbs but the in-form frontman saw his finish saved by goalkeeper Wes Foderingham just before the interval.
Groves introduced Josh Wakefield and Josh McQuoid at half-time, and Wakefield tested Foderingham with a curling effort.
At the other end, Roberts saw his volley drift wide before Jalal saved well from Collins.
Meanwhile, Steve Cook picked out Tubbs with a deep but the striker mistimed his effort.
Ritchie made sure Swindon claimed all three points as his cross crept in at the back post.
The Robins kept attacking and the woodwork stopped Williams scoring his second before Rooney came off the bench to add a late fourth.
Cherries: Jalal, Cook, Addison, Elphick, Daniels, Fogden (McQuoid, h/t), O’Kane, Partington, McDermott (Wakefield, h/t), Grabban, Tubbs.
Unused substitutes: Buchel, Zubar, Meades, Thomas, Fletcher.
Swindon: Foderingham, Thompson, Ward, McCormack, McEveley, Ritchie, Miller, Navarro (Ferry, 67), Roberts, Williams (Coke, 82), Collins (Rooney, 51).
Unused substitutes: Bedwell, Archibald-Henville, Benson, Bessone.
Goals: Ritchie (13, 68), Williams (25), Rooney (87).
Attendance: 8, 533 (752 Cherries supporters).