The defeat means that Nigel Clough's side are in 18th spot in the Championship table.
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Barnes scored twice in Derby's 4-2 League Cup defeat at Manchester United under new manager Nigel Clough.
And despite Kris Commons coming close to equalising, Nigel Clough's side fell to their third defeat in five matches.
The ex-Hull chairman has been at Derby since October 2007, and appointed Nigel Clough as manager in January 2009.
"It has been an absolute pleasure and privilege to work with Nigel Clough over the past 10 months, " he added.
And manager Nigel Clough has said his side will pick themselves up from Saturday's league defeat to take the game to Manchester United.
Brabin says he is full of respect for the way in which Burton have continued their promotion charge despite losing manager Nigel Clough to Derby.
Derby manager Nigel Clough was also upbeat after his side's late fightback that saw them put United under pressure after looking dead and buried at half-time.
The 23-year-old forward had been told he was not part of boss Nigel Clough's plans for the new season after failing to make the grade at Pride Park.
As rumours continue over who will be the new manager, Burton Albion's Chairman Ben Robinson told BBC Radio Shropshire he had not been approached by Shrewsbury Town to speak to Nigel Clough.
McFarland took over from Nigel Clough in January with the Brewers 13 points clear at the top but they could have finished second on the final day if Cambridge had beaten Altrincham by four goals.
Mrs Dickinson will be joining Brian Clough's son, Nigel, who is Derby County's current manager, at the racing circuit.
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