The striker struck with a superb 20-yard volley in the 79th minute - straight after the visitors thought they had rescued a point.
Dagenham rode their luck as Burton went close three times in the space of a minute early in the game. Kee headed against a post in the 12th minute before Chris Lewington made a double save to deny Calvin Zola and Robbie Weir.
Burton were well on top until 'keeper Dean Lyness, making his full debut, was adjudged to have tripped Jake Reed in the 28th minute. Lyness somewhat made amends as he pushed Michael Spillane's resulting spot-kick against a post but Dwight Gayle was on hand to tap in the loose ball.
The home side, however, turned the game around with two goals in three minutes early in the second half. Former Dagger Damien McCrory's deflected shot looped over Lewington from the edge of the area in the 51st minute, and then substitute Matt Paterson pounced on a mistake by Luke Wilkinson to slide the ball home.
Substitute Sam Williams put the Daggers level with a fierce right-footed shot in the 78th minute only for Kee to strike the winner for the hosts a minute later.