The Robins took advantage of Lee Tomlin's early dismissal to record only their sixth win of the season and move off the bottom, above both Barnsley and Posh.
Sam Baldock scored twice for City and both Paul Anderson and Stephen McManus were also on target.
McInnes said: "We owed our home fans a big performance and that's what we gave them.
"We were the better team apart from a five or ten minute spell after the sending-off, which seemed to affect us more than them.
"We were positive in all aspects of our play and it was a huge result for us. The sending-off made it difficult for Peterborough, but we were already on top at that point.
"It was a huge result for us and one we must use as a springboard. I'm certain we have what it takes to get out of trouble in the New Year."