The twice champion Brumbies had already clinched the Australian conference title but were looking for a victory at Lang Park to improve their seeding for the playoffs.
The narrow defeat left the Brumbies level with the Stormers on 34 points in the battle for the third seeding ahead of the South Africans' penultimate match of the season against Japan's Sunwolves on Saturday.
The Brumbies also lost flanker Chris Alcock, lock Rory Arnold and winger Aidan Toua to injury.
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Two penalties for Cooper to one for Wharenui Hawera had given the Reds a 6-3 lead after a forgettable first half most notable for Toua spilling a slippery ball with the line at his mercy.
The Brumbies went in front 11 minutes after the break when Reds flanker George Smith brought down a rolling maul that was rumbling towards the line and the referee awarded a penalty try.
Wallabies and Brumbies great Smith was sin-binned for his trouble and was still off the pitch when the Brumbies got their second try.
Cooper took the ball over his own try-line under pressure from Brumbies fullback Tom Banks but failed to ground it and Kyle Godwin pounced to score when it popped out of the flyhalf's grasp.
Replacement back Hamish Stewart replied with a converted try for the Reds nine minutes from time to bring the home side to within two points of the Brumbies at 15-13.
A turnover gave the Reds a last chance of victory and Cooper sent Izaia Perese charging for the line with a deft inside pass and the winning penalty was awarded for an infringement as the winger was dragged to the ground a few metres short.
(Reporting by Nick Mulvenney; Editing by Toby Chopra)
