Harrad scored twice in the last 10 minutes as the League Two club came from behind to record a famous third-round victory.
Gary O'Neil had marked his return to the Boro team with a fabulous free-kick to give the Championship club the lead just before the hour mark.
Boro made five changes from the draw against Norwich. Jason Steele, Barry Robson and Marvin Emnes missed out through injury, Tarmo Kink dropped down to the bench while speculation continued to surround the future of David Wheater.
In came Danny Coyne for his first game since the opening day of the season. Tony McMahon returned to right-back and Justin Hoyte moved to the right side of midfield, while Seb Hines replaced Wheater.
Kris Boyd came in for Kink and the other change made by Boro boss Tony Mowbray was the inclusion of O'Neil.
Burton, playing in the third round for the first time since famously taking Manchester United to a replay in 2006, made only one change as leading scorer Harrad replaced James Collins.
The two teams were separated by 45 league places prior to kick-off, but in an end-to-end first-half that was not immediately obvious.
Albion forced Boro back early on when a misdirected pass from Nick Bailey was seized upon, but Hines read the break well and cleared the danger.
At the other end, Boyd's free-kick was pushed away by Adam Legzdins.
Burton captain John McGrath tracked back from midfield to clear a goalbound effort from Julio Arca from just in front of his own line.
O'Neil then worked Legzdins again with a low shot from the edge of the area just before the break.
Boro stepped up a gear after the restart and after McMahon struck the crossbar with an 18-yard shot at the beginning of the second half, O'Neil put Mowbray's men ahead in the 58th minute.
O'Neil, making his first appearance of 2011 after illness, put the ball in the top left-hand corner from 25 yards out after Leroy Lita was brought down by Russell Penn.
Lita and Body wasted good chances when well-placed inside the area and Boro were made to pay as Harrad stole the headlines.
The striker, who is out of contract at the end of the season and looking for a move to a Championship club, suggested he might be able to cut it at that level with two tidy finishes.
The first came after 82 minutes when he turned home Penn's cross from 12 yards out.
Harrad then won the tie for Burton in the fourth-minute of injury time, smashing a shot past Coyne after Scott Malone's free-kick was deflected into his path.