Kee grabbed the 79th minute winner from the penalty spot after his first-half opener had been cancelled out by a Dean Winnard goal.
Goalkeeper Marcus Bettinelli kept Stanley on level terms with two fine reaction saves to deny Kee and Jimmy Phillips as Burton mounted an early onslaught.
The hosts were also frustrated when claims that a Robbie Weir header had crossed the line fell on deaf ears.
The visitors held out until Kee's 10-yard shot was deflected past the wrong-footed Bettinelli in the 34th minute after a cross by Phillips.
Stanley struggled to make chances and wasted a rare opportunity when Lee Molyneux blazed a 20-yard free kick too high in the 57th minute.
Molyneux was the provider with a free-kick that was glanced into the far corner by Winnard for Accrington's 69th minute equaliser.
But Burton were back in front 10 minutes later when Kee calmly stroked home a penalty after Phil Edwards had been dragged to the floor by Winnard.