After his side were beaten 3-1 at home in their second-round clash at the County Ground, the hot-headed Italian could be seen remonstrating with forward Clarke as the team left the pitch.
The former West Ham striker appeared to manhandle Clarke towards the tunnel and the pair seemed to be pushing and shoving before being separated.
Clarke refused to go into the dressing room and left the ground still wearing his kit while Di Canio went home without attending his usual post-match press conference.
Instead it was left to chairman Jeremy Wray to try to explain the scuffle.
Wray, who has promised to conduct a full investigation, said: "Clearly there was an incident and it would be wrong to apportion blame until we have all the facts.
"Leon and the fitness coach were having words after the game and there was a disagreement between them. As Leon came off the pitch Paolo was conscious these things should be done behind closed doors.
"There was a misunderstanding there that carried on into the tunnel.
"It got to a situation where the whole thing blew up very fast. There was frustration on both sides.
"The whole issue was trivial but nobody responded in the right way.
"The manager was quite right to try to defuse the situation.
"Paolo was frustrated the whole thing was played out in public. He feels one of the players has let him down."
The incident overshadowed another impressive performance from Southampton, who continued their impressive start to the season to set up a third-round showdown with either Charlton or Preston.
Brazilian Guly do Prado gave the visitors the perfect start by heading in Lee Holmes' cross after only 17 minutes.
Jonathan Forte's neat finish from 12 yards on the half-hour - the former Scunthorpe striker's first goal of the season - doubled Saints' lead.
Swindon substitute Mehdi Kerrouche's late free-kick threatened to set up a grand slam finish but Rickie Lambert sealed the win in the dying seconds.
While all the talk was about Di Canio and Clarke, Saints manager Nigel Adkins was just pleased to get through.
Adkins said: "We passed the ball around really well in the first half but fair play to Swindon, they made a game of it in the second.
"We try to play football in the right way and I was very pleased with another solid performance and we look forward to the next round."