The Blades got off to the perfect start with a goal inside five minutes as Danny Webber skipped past Burnley left-back Graham Branch and crossed low into the near post area where fellow striker Shipperley converted easily.
The second goal five minutes before the interval was a bizarre affair as Michael Tonge's corner was initially turned towards goal by the head of Morgan, only for Burnley keeper Brian Jensen to fist the ball against the shoulder of the big defender and into the net.
Burnley's consolation came 18 minutes from time when Ade Akinbiyi deflected a shot from Gifton Noel-Williams past Paddy Kenny.