Kay nodded in Shaun Williams' 87th-minute free-kick, moments after Blades substitute Febian Brandy, who was later sent off, missed an open goal.
Brandy, a 63rd-minute replacement for Jamie Murphy, inexplicably steered his 85th-minute effort wide before seeing red for a challenge on Dons defender Lee Hodson.
Robinson, whose side remain unbeaten in League One, said: "It was smash-and-grab. I thought for 60 minutes it was a stalemate with two teams whose two styles were so similar.
"It was like, we try and play, you try and play, but we knew we could stop them. They knew they could stop us and it wasn't until the substitutions that the game opened up.
"I think Febian's miss was a big moment in the game and obviously we hit them with a set-piece; it's a very rare way we win games.
"There's a tremendous togetherness among the group and this is a special place to visit in League One. It's always an honour to come and play here - and to walk away with three points, it's something we will remember for a long, long time.
"To say you've come to Bramall Lane and got three points, there're not many managers at this level of English football who can say that."