A howler from Blades keeper Paddy Kenny gifted Cardiff City an equaliser and they went on to claim a deserved victory.
There looked little danger in first-half injury-time when Richard Langley hit a free-kick from 35 yards, but Kenny somehow fumbled the ball into the net off his near post.
United had taken the lead when Peter Ndlovu crashed a shot into the net when the Bluebirds defence failed to clear a 25th minute corner.
Both sides went for the win in the second half but it was the home side who scored the decisive goal in the 63rd minute.
Once again it was a cross into the box that caused the trouble and when the ball was cleared it went straight to John Robinson who steered the ball back across Kenny and into the corner of the net for his first goal for the club.
City boss Lennie Lawrence was delighted with his side's performance.
"I'm very proud of the way we played as they are a difficult side to beat and I am pleased that we came from a goal behind to win," he said.
"We showed a great defensive quality and were superior in both penalty areas. Our second goal was quality and overall we shaded it.
"They have beaten us twice this season but it was our turn tonight. I'm not thinking about reaching the play-offs but there are no easy games at this club and we will keep going until the end of the season and I will see then where we finish." Blades manager Neil Warnock was very unhappy about his side's defeat.
"We didn't deserve to lose but goals change games," he moaned. "I was very disappointed today as the Cardiff players seemed to get away with everything while we were always pulled up.
"Even so, we had chances in the second half just before they scored.
"Our keeper made a mistake but these things happen and it has cost us dearly. That equaliser gave them a lift and it was a kick in the teeth for us right on half-time."