Defender Kay powered in Shaun Williams' 87th-minute free-kick, moments after Blades substitute Febian Brandy, who was later sent off, missed an open goal.
Dons striker Izale McLeod had the best opportunity of the opening stages but the former United player skewed his 14th-minute effort wide after tricking his way past defender Harry Maguire.
The home side responded and debutant Jose Baxter thumped a volley the wrong side of the right-hand upright.
United should have snatched the advantage minutes into the second half but striker Lyle Taylor squandered the opportunity to grab his maiden Blades goal by diverting Baxter's deflected effort narrowly wide.
Dons goalkeeper David Martin dived at full stretch to turn away Tony McMahon's free-kick before Brandy inexplicably fired wide of the gaping goal from Baxter's centre five minutes from time.
Kay then nodded home at the back post and United's misery was compounded in injury time when Brandy received a straight red card for a late tackle on Lee Hodson.