City fielded new signing Martin John at left back and Jason Koumas was amongst the subs.
City took the lead when Peter Whittingham was allowed too much time and space on the left and his square pass found Bothroyd to make it 1 - 0.
City's joy didn't last long as Burton levelled a few minutes later. Some defensive hesitation allowed Harrad to go through on goal a clip the ball over Tom Heaton in the City goal.
Jay Bothroyd appeared to take a knock and was replaced by Ross McCormack at half time. Adam Matthews left the field in some discomfort to be replaced by Paul Quinn and then Gavin Rae made way for the arrival of Jason Koumas.
New signing Martin John was clearly injured and 'walking wounded' but he kept on playing as the game headed towards 90 mins.
McCormack scored with a header that was ruled out for offisde and with the last action of the 90 mins Burton scooped the ball over the bar with the goal gaping.
The opening period of extra time remained goalless but within 2 minutes of the start of the 2nd period of extra time City struck. Ross McCormack was played in by Koumas and City were 2 - 1 up after the ball went in off the post.
Just a couple of minutes later and it was 3 - 1. Tom Heaton punted upfield, Ross controlled, turned, made space and rifled home his second and put the game beyond doubt.
City weren't finished though and Koumas again claimed an assist as he played Michael Chopra through on goal and the City marksman made no mistake.
A harsh scoreline on Burton but some clinical finishing in the final minutes made it look a far more comfortable result than the performace deserved.
McCormack ran to the Ninian Stand at the final whistle and handed his shirt to a fan. Hard to say if his waves were ones of farewell.
After the final whistle we learned that we are away at Peterborough in round 2.
Next up City are away at Derby on Saturday.