The home side almost got off to the perfect start when Paul Coutts raided down the right and fired a cross towards the near post which somehow struck the woodwork, and the visitors cleared their lines.
The hosts took the lead when Sammon came down the left and squared the ball for Craig Bryson in the middle, who made no mistake firing in from close range.
After the break, Nigel Clough's side got their second when Chris Martin turned his man on the edge of the box and fired low and hard into the bottom corner of the net.
It was three soon after when Jamie Ward's corner was flicked into the path of Richard Keogh at the back post, heading the ball into the net.
Peterborough were offered a way back into the game when Frank Fielding brought down Dwight Gayle in the area and he was duly sent off. Grant McCann dispatched the resulting penalty past sub keeper Adam Legzdins.
Lee Tomlin could have got his side back in the game when the ball bounced kindly for him in the Derby area but he snatched at his shot and it went wide of the post.
Peterborough pushed to get back into the game but couldn't find another breakthrough and Derby were able to see out the win.