The striker scored one and created another, while his pace was a constant menace to the Burnley defence all afternoon as United recorded their 10th victory in 12 home games this season.
The Blades had the visitors pegged back in their own third for most of the first half and after a slow start began to create chances.
Webber looked sharp from the start and had the first clear opening when the ball sat up kindly for him and he struck a half-volley which Burnley goalkeeper Brian Jensen held at his near post.
Alan Quinn then saw a good effort deflected over by Micah Hyde before Jensen was lucky that Leigh Bromby's close-range header was straight at him.
Burnley immediately responded as Gareth O'Connor played in Ade Akinbiyi and the striker's low, angled shot was tipped round the post by Paddy Kenny.
Shipperley then found himself through when Jon McGreal missed an interception, but the former Crystal Palace striker's shot went just wide as Jensen closed him down.
The breakthrough came in the 29th minute when Webber ran into the box where Shipperley took the ball off him and struck a powerful shot into the top right-hand corner of the goal.
Burnley almost replied immediately as McGreal got up to head a corner goalwards, forcing a good save from Kenny.
United then doubled their lead. Montgomery's deep cross was headed back across goal by Craig Short and Webber was on hand to head in at the far post.
Burnley began the second half with more purpose. John Spicer tried his luck from range and then fired a free-kick into the arms of Kenny, and O'Connor also found space to shoot but could not beat the United goalkeeper.
But manager Steve Cotterill was still not happy with what he saw and in the 59th minute he made a double substitution, bringing on Graham Branch and Gifton Noel-Williams for O'Connor and Spicer. Almost immediately, Akinbiyi laid the ball off for Branch to shoot from outside the area, but Kenny easily gathered his tame effort.
United dropped back but were still creating chances on the break. Webber was close to a second goal when Quinn fired in a low cross, and Jensen had to make a smart save at his near post. In the 69th minute, Noel-Williams beat the offside trap to break into the box but after he checked back to create space to shoot, his effort was charged down.
Akinbiyi was having a frustrating afternoon, summed up in the 75th minute when Bromby missed an interception and the ball fell to him in front of goal. As he shaped to shoot, Bromby stretched, getting a touch which sent the ball over for a goalkick.
United then wrapped things up in the 81st minute when Chris Armstrong - making his first start of the season after long-running injury problems - crossed from the left to Shipperley at the far post.
He had time to bring it down before firing a low shot back across goal and in at the far post.