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Manchester United loanee Nick Powell reduced the deficit to two shortly after the break, but the strike proved little more than a consolation as promotion-chasing Derby moved into the playoff positions in England's second tier.
McClaren watched on from the dugout as his side took a sixth-minute lead through Bryson, whose 20-yard free kick proved too good for goalkeeper Lee Nicholls.
It soon got worse for the Latics as the Rams doubled and trebled their advantage shortly after, first through Dawkins, who smashed home the second, and then Martin nonchalantly stroked home the third from the penalty spot on the half-hour mark.
Powell was on hand to reduce the arrears five minutes at the interval, but Owen Coyle's men failed to build the momentum necessary to stage a comeback and Derby cruised to the three points.