"When you go two up so quickly you hope to go on and win the game, but after conceding two we still had enough about us to recover and take the lead again," said Robins.
"We are not the finished article. I had Kieran Trippier playing wide in midfield instead of in his usual right-back position, yet he still caught the eye and played well. That applied to several of our new players and I am sure we will improve as we go along."
Andy Gray bagged two goals inside the opening 12 minutes to put the Tykes in control, before Marvin Elliott and Albert Adomah pulled City level.
Adam Hammill put Barnsley in front once again early in the second half and it took a 62nd-minute penalty from David Clarkson to earn the home side a point.
Robins added: "This time last season we had one point and that was still the case after seven games so I'm pleased with the way things are developing.
Bristol City: James, Hunt, Stewart, Fontaine, McAllister, Elliott, Skuse (Johnson 10), Campbell-Ryce (Akinde 73), Sproule, Clarkson (Gavin Williams 89), Adomah. Unused substitutes: Gerken, Carey, Jackson, Tom Williams.
Barnsley: Steele, Hassell, Foster, Shackell, McEveley, Trippier (Morales Neumann 88), Arismendi (Doyle 82), Colace, Hammill, Lovre, Gray. Unused substitutes: Preece, Hume, Devaney, Butterfield, Bennett.
Referee: Andy Penn (W Midlands).
Man of the match: Andy Gray.
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