The chance to go top of the Championship was taken away from them after they lost at home to Rotherham.
The Millers captain coolly turned the ball home after Chiedozie Ogbene found him in the box.
Enda Stevens had the ball in the back of the net before the start of the second half, but it was ruled out for an obstruction.
Billy Sharp headed wide for a Blades side that failed to take anything from a game they dominated.
If the Blades win this South Yorkshire derby, they will move above the Clarets in the table.
After just 20 minutes, the hosts had to make an early change due to an injury to John Fleck.
The Millers, who had six first-team players missing through injury, were the first team to threaten, with Wes Foderingham tipping a shot from Peter Kioso over the bar.
Wiles scored the only goal of the game in the 38th minute after Jack Robinson failed to clear the ball, and United struggled to create many chances of their own.
Stevens, who had a goal disallowed late in the first half, appeared fortunate not to give away a penalty deep into injury time when he tangled with Wiles.
The Blades came out strong in the second half and were able to get a goal when McBurnie headed in a corner in the 69th minute.
Last week, the Millers were stunned by two last-minute goals, including a 100th-minute winner, but they refused to give up against the third-place Blades.
Paul Heckingbottom is the manager of the United Kingdom'sSheffield United.
We did not play with the same pace and energy that we usually do.
You don't want to concede the first goal when you have a team that feels that. There are likely reasons why we fell below our levels. We weren't as good as we could have been.
We did not win that game, but we could have. I don't think we deserved to win, but we didn't play with the same gusto. As a group, we weren't at our best.
Matt Taylor is the boss of the club.
I told the team before the game what it would be like for the fans to watch it.
"I used myself as an example because I am a Man Utd fan and the games I remember growing up are United- City, United-Liverpool, United-Leeds, the ones with a bit more spice about them."
The local derbies meant a lot to the fans.
It means a lot to the players because they have taken a few blows recently.
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