Goals from Craig Bryson, Chris Martin and Richard Keogh saw the Rams take control and despite Grant McCann converting a penalty for Posh after goalkeeper Frank Fielding was sent off, they held on for a comfortable victory.
"We were totally in command as soon as the first whistle went," said Garner.
"The first 20 minutes, we were absolutely superb, dominated the game and were in their half constantly.
"We didn't get the goal our play justified but we stuck at it and when it came, it was a brilliant goal, a training ground goal really.
"Great hold up play from Conor Sammon and a typical Craig Bryson run and a good finish."
Garner had no complaints with Fielding's red card and admitted the keeper was at fault.
"At the time I thought his starting position was too deep and I think he hesitated which was disappointing," he added.
"The last thing we wanted was a penalty and a sending off. It was a poor decision by Frank, not the referee, who got it spot on."