Hull City's perfect home record came to an end as they lost at the MKM Stadium.
McBurnie scored his third goal in as many games when his 20th-minute drive went through the hands of the goalkeeper and into the net to give the Blades a 2-0 lead.
In-form Oscar Estupinan was denied twice by Wes Foderingham after he was sent clean through before Ozan Tufan volleyed on to the roof of the net as Hull failed to make the most of their chances tolevel.
The Blades made sure of the points in the second half when Berge's shot came back off the underside of the bar.
They returned to the top of the table after their rare away win, their second in their past 11 league matches on the road. The last nine times the Blades have traveled to Hull, they've won.
For the first time in more than two decades, the Tigers started a season with four straight home victories. They slipped to 12th place after a second-straight defeat.
Despite facing some vociferous support from a crowd of more than 20,000 - Hull's biggest at home for five years - it was the Blades who settled quicker with Berge whistling a 30-yarder wide before Tommy Doyle's shot on the turn drew a smart save from the keeper.
Benjamin Tetteh was forced to leave the game early with a injury after stretching to reach the ball.
They fell behind when McBurnie, moments after failing to connect with George Baldock's centre, picked up the ball on the left of the box and smashed it into the net.
The Scot has gone 43 games without scoring but has three goals in as many games.
Estupinan had scored all five of Hull's goals in the last two games.
The Colombian had a chance to equalize after 25 minutes when he raced clear from the halfway line following a Blades corner, but Foderingham saved both of his shots.
After Estupinan sent Tufan through, the Turkey international swooped on the ball to the roof of Foderingham's net, only to be denied by the woodwork.
The result was settled by the excellent Berge after the excellent Ryan Longman and Iliman Ndiaye both hit the woodwork.
Ndiaye scored his third goal of the season when his shot from just inside the box nicked off Jones and flew into the net.
Even with the help of nine minutes of injury time, they were unable to avoid a first home defeat of the season.
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