Dons striker Ryan Lowe scored a brace to cancel out Tom Eaves' early double, but Kay needlessly volleyed past his own goalkeeper in the 67th minute to gift the Shrews their first win in four games.
Shrewsbury raced into a shock 2-0 lead inside the first 19 minutes as Eaves shrugged off Shaun Williams to drill in the opener before neatly curling in his second of the game from the edge of the box.
But Lowe pulled a goal back for the Dons in the 27th minute, bundling in the ball from close range after Kay had flicked on Dean Bowditch's corner at the near post.
Lowe equalised for the Dons two minutes after the break when he headed in Williams' inswinging free-kick at the far post.
Izale Mcleod then blazed a good chance wide for the home side before Kay volleyed Matt Richards' hopeful long pass past his own keeper Ian McLoughlin to gift Shrewsbury the lead.
The Dons piled forward in the dying minutes but Shrews keeper Chris Weale superbly denied Bowditch and substitute Alan Smith to condemn the hosts to a first defeat in seven games.