Posh arrived at Pride Park as the division's form team having set a club record by going 10 games without defeat at this level but they struggled to live with the Rams until home goalkeeper Frank Fielding was sent off in the 62nd minute.
Substitute Grant McCann scored the resulting penalty after Fielding's dismissal but Darren Ferguson's side had fallen behind to goals from Craig Bryson, Chris Martin and Richard Keogh either side of half-time.
Huddersfield's win over Millwall earlier on Saturday saw Peterborough drop a place to 21st before kick-off and they are now only outside the relegation zone on goal difference.
Former Posh playmaker Paul Coutts' mishit cross that smashed against a post after two minutes put the visitors on the back foot and that is where they remained for the entire first half.
Martin, Connor Sammon, Craig Forsyth, Keogh and Coutts all missed further chances, but the hosts finally got the goal they deserved five minutes before half-time when Ward and Sammon combined well down the left and the latter picked out Bryson in the box to blast the ball home from close range.
Visiting boss Ferguson made two changes at the break, introducing McCann and Nathaniel Mendez-Laing for Gaby Zakuani and Kane Ferdinand.
Perhaps Peterborough were still trying to reorganise themselves at the start of the second half as, in the 52nd minute, Martin doubled Derby's lead with a superbly-taken powerful, low shot on the turn and then Keogh headed in a third goal when he was left unmarked at the far post after Forsyth had flicked on Jamie Ward's corner.
The Rams were cruising but they suffered a setback just after the hour mark when Fielding brought down Dwight Gayle, who was clean through on goal.
Fielding received a straight red card before McCann sent his penalty past substitute keeper Adam Legzdins, who was brought on at the expense of Martin.
Lee Tomlin had a good chance to pull another goal back after 73 minutes but he dragged his shot wide from 12 yards out.
Gayle then saw a shot blocked as Legzdins was called upon to make a save for the first time.
Sammon had a chance to finish off the game in the second of seven minutes of second-half stoppage time but he was denied by Bobby Olejnik.