A lengthy stoppage, as Adam McGurk was stretchered off the pitch, meant it was a late break for both sides.
At the other end Fleetwood midfielder Gareth Evans rattled the post, having left two Burton defenders in his wake.
The game looked to be heading to a stalemate until a snap-shot from Jeff Hughes forced a corner.
From the subsequent set piece, Northern Ireland international Hughes surged through the crowded area and shot low past Lyness.
Town looked like they were going to hang on, but in time added on Phil Edwards flung in a long throw from the right and the ball bounced kindly for Kee to turn and fire past Davies.