The Blues had responded to their 3-2 defeat to West Ham on July 1 with back-to-back wins over Watford and Crystal Palace but were easily second best in South Yorkshire.
The visitors dominated possession but did nothing with it in a dismal display, their misery compounded as McGoldrick doubled his tally 13 minutes from time.
United leapfrog Wolves into sixth place, having played a game more. Chelsea are six points ahead in third but remain only a point above fourth-place Leicester City and two ahead of Manchester United in fifth.
McGoldrick ended his wait for a top-flight goal in the simplest of fashions as he hammered home the rebound from point-blank range after Kepa Arrizabalaga had reacted well to McBurnie’s deflected volley.
Fifteen minutes later, McBurnie was not to be denied as he headed Enda Stevens’ left-wing cross into the bottom-left corner after the Blades cut through a lackadaisical Chelsea defence.
Dean Henderson did well to preserve the two-goal advantage, getting down low to keep out Reece James’ drive.
Chelsea offered precious little as an attacking force despite controlling the ball, with the introduction of Olivier Giroud in the 66th minute failing to spark a revival.
McGoldrick made sure there would be no comeback as he pounced on a dreadful error from Antonio Rudiger, with Lys Mousset spurning a gilt-edged chance to make it 4-0 in a rare mistake from a home side excellently positioned to play at least in the Europa League for the first time.