Lee Tomlin was sent off for Posh after just 11 minutes, but City were already ahead through Paul Anderson's goal.
Grant McCann equalised for the visitors before Sam Baldock netted a double either side of Stephen McManus' header.
And not even Dwight Gayle's late consolation could dampen McInnes' mood.
"We owed our home fans a big performance and that's what we gave them," he said.
"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."
- Sports & Recreation
- Sam Baldock