The visitors were bidding to equal the all-time English top-flight record of 18 successive wins as well as the best-ever Premier League start of nine wins from nine, but in the end they will be relieved to have come away with a point following an uncharacteristically poor display.
United rose to the occasion despite going into the match having won just twice in their last 13 league games, and Marcus Rashford gave them the lead in controversial circumstances nine minutes before the end of the first half.
VAR was again the talking point at half time after allowing that goal to stand and denying Liverpool an equaliser through Sadio Mane, and for much of the contest it looked as though Liverpool would fall to just their second defeat in 48 Premier League games.
Adam Lallana's equaliser five minutes from time rescued a point for the visitors late on, though, leaving them six clear of champions Manchester City despite failing to win in the league for the first time since March 3.
The team news from both camps was significant as goalkeepers David de Gea and Alisson Becker were passed fit to start, whereas Mohamed Salah failed to recover from his ankle injury in time as Divock Origi came into the side.
The unfamiliar system caused Liverpool problems throughout the first half as the league leaders struggled to produce anything like the football which had seen them win their last 17 top-flight matches, although clear chances were few and far between at either end.
Tame long-range efforts from Georginio Wijnaldum and Scott McTominay were the only shots of any note in the opening 30 minutes, although it was United who would have been happier with their performance.
Nonetheless it was Liverpool who created the first big chance of the contest when Mane finally found some space down the right before playing a low pass inside for Roberto Firmino, whose finish made De Gea's save easier than it should have been.
Just two minutes later United took the lead as Rashford finished off a Daniel James cross at the end of a devastating counter-attack, although it was the events in the build-up to that which courted controversy.
Liverpool felt there was a foul on Origi when United won the ball back and the decision was checked by VAR, which upheld Martin Atkinson's on-field call to let play carry on.
The visitors' frustration with VAR was increased just six minutes later when they had a goal of their own disallowed, although on this occasion the ruling was clearer as the ball hit the hand of Mane while he controlled it before holding off Victor Lindelof and firing past De Gea.
Regardless, there was a general feeling of injustice among the Liverpool players as they went in for half time, although they could not have too many complaints at being behind following an uncharacteristically nervy display.
It left Jurgen Klopp's side needing to become the first visiting team since May 1984 to overcome a half-time deficit and go on to win a league game at Old Trafford, and there was an initial improvement after the interval as they began to see more of the ball.
It was not until the 64th minute that the first shot of the second half arrived, though, and even then it was a hooked strike from Alex Oxlade-Chamberlain which ended up comfortably wide of the target.
United's pace on the counter was always a lingering threat and almost brought about the second goal which has proved so elusive in recent weeks when Rashford fired narrowly wide.
Fred also had two sights of goal, the second of which came closest, but United were once again made to pay for their inability to kill a game off when Liverpool levelled things up five minutes from time.
It had been difficult to see where an equaliser would come from until one of many poor crosses, this time from Andrew Robertson, found its way all the way through to the back post, where Lallana was waiting to tuck it home.
The goal entirely changed the atmosphere inside the stadium and the momentum on the pitch, with Liverpool suddenly looking most like scoring as Trent Alexander-Arnold and Oxlade-Chamberlain both came close.
United held on throughout five minutes of stoppage time to pick up a draw which lifts them up a place to 13th in the Premier League, having made their worst start to a season since 1986-87.
The Reds are still winless at Old Trafford under Klopp and have only won one of their last 11 league meetings with their biggest rivals, with five of the last seven ending as draws.
However, they remain unbeaten this season and in control of the Premier League title race after almost a quarter of the season, extending their undefeated streak to 26 games stretching back to January 3.
MAN UTD (3-4-1-2): De Gea; Lindelof, Maguire, Rojo; Wan-Bissaka, McTominay, Fred, Young; Pereira (Williams 94'); James, Rashford (Martial 84')
LIVERPOOL (4-3-3): Alisson; Alexander-Arnold, Matip, Van Dijk, Robertson; Henderson (Lallana 71'), Fabinho, Wijnaldum (Keita 82'); Mane, Firmino, Origi (Oxlade-Chamberlain 60')