Dundee United led through John Souttar's 51st minute goal but Barry Robson's penelty and Peter Pawlett's last-minute header sealed a 2-1 success.
"In the second half we created so many chances and their keeper's pulled off three or four great saves," McInnes said. "To come from behind and win it, I think we got what we deserved.
"There was two teams desperate to win the game and I thought, generally, the referee handled the game well. It was competitive and aggressive but I don't think it ever boiled over at any point.
"We wanted to come here and win and I told the players at half-time to demand more of themselves. These boys never know they're beaten and always try to win the game. Peter Pawlett was outstanding today."
Jackie McNamara's side were suffering their third consecutive defeat, and the United manager said: "We can play a lot better and I don't think it was a very entertaining game; very stop-start.
"There were a lot of free-kicks in the game and niggly fouls. It's a good learning curve for our boys."