Manchester suffered a second consecutive defeat in the Premier League after losing 2-0 to Burnley at Old Trafford on Wednesday.
After a 2-0 loss at Anfield on Sunday, Chris Wood and Jay Rodriguez scored in each half as United failed to close the gap on the top four.
Ole Gunnar Solskjaer's side remain six points behind Chelsea while Burnley climb to 13th place after back-to-back wins.
The hosts enjoyed the best chances in the first half but Anthony Martial and Juan Mata failed to convert from two Aaron Wan-Bissaka crosses.
Burnley took advantage of United's wastefulness after 38 minutes when Ben Mee flicked on Ashley Westwood's free kick for Wood to smash home.
After the break, Burnley scored a second after 56 minutes when Rodriguez produced a stunning strike into the top corner.
Luke Shaw had a late goal ruled out as United struggled to create any clear cut opportunity after Burnley's second.