But, Solskjaer decided to keep youngster Mason Greenwood on the bench, and play without a recognised striker.
Chris Wood opened the scoring for Burnley in the first half, before Jay Rodriguez confirmed the win.
Rodriguez managed to beat David de Gea at his near post from 20 yards, which condemned United to their third defeat in four matches.
Next up for the Red Devils is a trip to either Watford or Tranmere Rovers in the FA Cup, before heading to the Etihad Stadium to take on Manchester City in the Carabao Cup.