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Bruno Fernandes fired United into the lead in the 10th minute before Diogo Jota pulled back an equaliser for Liverpool in the 34th minute.
Roberto Firmino handed Liverpool the advantage going into half-time by adding his side's second.
Marcus Rashford pulled one back for United to hand his side a fighting chance, but Mohamed Salah scored late on at Old Trafford to seal a sensational win for Liverpool.
Liverpool's victory on Thursday ensured three crucial points as the Reds push for a Premier League top-four spot.
United, on the other hand, have slipped to back-to-back defeats and sit in second place behind newly-crowned Premier League champions Manchester City.
Keane pulled no punches towards McTominay and Fred as he warned that United "will not be winning any big trophies" with the duo in midfield.
The 49-year-old Irishman told Sky Sports: "All of United's shortcomings have been shown up in the last couple of days.
"Look at the performance of the two midfielders. McTominay is a good, honest player. And they've got Fred. As long as those two players are in midfield for Manchester United, they will not be winning any big trophies.
"I can't believe how short Manchester United are."
Keane continued his ruthless criticism of United's poor performance, saying there was "mistake after mistake" and "there wasn't enough quality" in the squad.
The United legend explained: "I know Ole perseveres with Fred, but I can't see it. How they think Fred is going to get Man Utd back to winning titles is beyond me.
"The goalkeeper didn't look great tonight either. There was mistake after mistake. Poor, poor night for United. [Harry] Maguire is a loss, but these are schoolboy errors... It's nothing to do with missing Maguire.
"The performance tonight was nowhere near good enough. All of United's shortcomings have been shown up in the last couple of days. Tonight, there wasn't enough quality. Liverpool could have scored six or seven.
"This squad is so short of competing against Man City to win the title. Man City are so far ahead of this squad it is scary.
"I think Ole will reflect on the last few days and think, 'We need three or four big players to come into this club this summer.'"
Keane also had strong choice of words towards Fernandes as he accused the United star of spending "half the night crying on the pitch."
Featured Image Credit: PA/Sky Sports
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