Jurgen Klopp's side came out the wrong side of a seven-goal thriller despite taking the lead on two occasions at Old Trafford.
After falling behind in the 10th minute, the Reds netted two goals in quick succession to take a narrow lead into the half-time interval.
Alexis Mac Allister restored parity in the 44th minute, before Salah latched onto Andre Onana's parry to fire his right-foot shot in off the post.
Although Antony netted a late leveller, Liverpool appeared to be on their way to Wembley when Harvey Elliott restored their advantage in the first half of extra time.
However, their hopes of winning their second FA Cup of the Klopp era were quashed by a Marcus Rashford equaliser and Amad Diallo's 121st-minute equaliser.
While it proved to be a disappointing result for the Reds, Salah still managed to create a piece of history by becoming the first player to score in five successive away matches at Old Trafford.
The Egyptian began that goalscoring run with a brace in an FA Cup fourth-round defeat in January 2021, before he found the net in the Premier League clash in May of the same season.
Salah then scored one of his three Premier League hat-tricks in a dominant 5-0 win over the Red Devils in October 2021.
The forward continued his success at Old Trafford with a strike in a 2-1 defeat in Liverpool's most recent Premier League away game against Man United in August 2022.
After going on to make history with his effort on Sunday, Salah will have the chance to continue his impressive goalscoring run when the Reds return to Old Trafford for a Premier League fixture on April 7.
In total,
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