Kee's 78th-minute header clinched the hosts a second home win in four days - and sentenced the Stags to a fifth successive defeat.
The Stags had an early scare when goalkeeper Alan Marriott had to get his fingertips to a Chris Hussey free-kick that found its way through a crowded area.
Marriott also had to produce a smart save to deny Dominic Knowles as Burton pressed again in the 19th minute.
The visitors had a chance to snatch the lead in a rare attack a minute before the break, but Lee Stevenson's shot was superbly kept out by Dean Lyness.
The Burton keeper also saved well with this legs when substitute Lindon Meikle rose for a far post header in the 57th minute and came to the rescue again in the 73rd minute when he got down to smother a shot from the same player as the visitors sensed victory.
But it was the hosts who went in front five minutes later when Kee arrived at the far post to nod in Adam McGurk's cross.