Winger Matt Ritchie opened the scoring before providing for Tommy Miller and Raffaele De Vita as Barker's side failed to cope with their opponents.
The Red Devils followed the Robins out of League Two last season but Barker feels the step up to League One is going to test his squad over the course of the season.
Barker: "For the club it is a big challenge. What we have to do is learn to adapt quickly to the rise in class and deal with the sort of players we are now going to meet every week.
"We have to look forward to the year ahead and enjoy playing in the better stadiums and on the better pitches you find in League One."
Barker added: "We knew from last year that Swindon were a good side and they have strengthened considerably over the summer. They have big ambitions."