A goalless stalemate stretching back to the first meeting at London Road 10 days ago was broken by the former Notts County striker in the 73rd minute.
The goal earned Burton the fourth League scalp in their history and a home tie against Burscough in the next round.
Posh dominated the first half but Burton worked hard to limit their scoring opportunities.
Midfielder Adam Newton had a 12-yard header saved by Saul Deeney in the 12th minute and the Burton keeper also dealt with a shot from Jamie Day.
Danny Crow set up the first clear-cut chance for the visitors in the 27th minute but Newton's shot from the edge of the area was tipped around the post by Deeney.
Posh keeper Mark Tyler was more or less a spectator in the first half but Burton came out after the break with much more attacking ambition.
Harrad gave early warning of their intentions with a low shot that flew just wide from the edge of the area and skipper Darren Stride saw two headers miss the target.
It was Stride who made a vital contribution to the deciding goal when he hooked the ball into the area for Harrad to hold off Chris Plummer before firing home a six-yard shot.
Posh had one chance to take the tie into extra time but Day's low shot in the last minute was well saved by Deeney.