Ole Gunnar Solskjaer was forced to make all three of his substitutions in the first half, but Liverpool could not capitalise on that disruption to take home all three points.
The Reds do still regain top spot from Manchester City, but may feel that they have missed an opportunity to gain real momentum in the Premier League title race.
It was a nervy start from United with Ashley Young's short first-minute back-pass forcing David de Gea to handle the ball under pressure from Roberto Firmino, but James Milner hit the direct free kick from inside the box straight into the wall.
The story of the first half was the spate of injuries which hampered both sides, United in particular. Marcus Rashford appeared to be the first casualty but played through the pain while his team lost Ander Herrera and Juan Mata inside the opening 24 minutes.
United were already without Nemanja Matic - who missed out altogether with a muscle strain - when his midfield partner was forced off, and Mata pulled up moments later, but not before a crucial final contribution to block Mohamed Salah as he ran into the box.
Solskjaer brought on Andreas Pereira and Jesse Lingard, with the latter himself only 10 days removed from a hamstring strain picked up in the 2-0 Champions League defeat to Paris Saint-Germain, during which United were again hit by first-half injuries.
Liverpool then suffered a setback of their own as Firmino twisted his ankle leading to the early introduction of Daniel Sturridge, who immediately tried to catch out De Gea with a 30-yard sighter which bounced comfortably into the United goalkeeper's hands.
There were more substitutions than shots on target in the opening 45 minutes, but Paul Pogba almost came up with a moment of quality, shaping a curling 25-yard strike towards the far corner which had to be blocked by the head of Virgil van Dijk.
United were in the ascendancy before the break and had the only clear-cut chance of the half when Romelu Lukaku picked out the run of Lingard, whose attempt to round Alisson and slot into the net from eight yards was expertly thwarted by the Brazilian.
Lingard came away from that incident having aggravated his hamstring injury, and Solskjaer was forced to bring on Alexis Sanchez for his third and final change on 43 minutes, while Rashford was still a concern, seemingly inhibited by his earlier issue.
Lukaku tried to take the goalscoring onus on his shoulders but sent his back-post header straight at Alisson on the stroke of the interval.
Solskjaer took the opportunity of the half-time interval to regroup his players, who came out in a 4-5-1 shape with the intent of sitting back and hitting Liverpool on the counter-attack - and yet they still created the best chances to break the deadlock.
Rashford dragged a snap-shot past the post on 56 minutes and Pogba and Sanchez both got in beyond the Liverpool backline from free kicks only to send headers wide, albeit from offside positions as the division's best away defence remained organised.
The Reds were far from their incisive best with the 64% possession that they did have, with Sadio Mane guilty several times of wasting promising positions and Salah shackled by Luke Shaw, so much so that he was hooked with 10 minutes to go when Liverpool were still chasing a winner.
Georginio Wijnaldum and Joel Matip headed off target from back-to-back Liverpool corners on 70 minutes, with both appealing that they were impeded by Scott McTominay, but referee Michael Oliver felt that the penalty-box wrestling was within the rules.
From a set piece at the other end, Matip was relieved to see the flag go up against Chris Smalling, whose knock across the box was clumsily bundled into his own net by the Cameroonian defender on 75 minutes, the closest that the match came to a goal.
Jurgen Klopp introduced Xherdan Shaqiri for the final quarter of the game, hoping for a repeat of the impact that the Swiss made in his two-goal cameo in December's reverse fixture, but even he could not improve a sub-standard attacking display from the visitors.
In the latter stages it became apparent that both managers were content with a share of the spoils, but Smalling could have won it for United in stoppage time had he not hesitated when trying to get on the end of a inswinging cross to the back post by Lukaku.
For the first time in the Premier League era, a meeting between these two sides at Old Trafford finished goalless, the result a triumph of the togetherness instilled by Solskjaer into the Red Devils, who, though depleted, were roared over the line by a raucous home crowd.
Arsenal's victory over Southampton means that United do drop out of the top four and their Champions League hopes may hinge on the extent of their injury problems. Meanwhile, Liverpool have a one-point advantage over champions City heading into the run-in.
MAN UTD (4-4-2): De Gea; Young, Lindelof, Smalling, Shaw; McTominay, Herrera (Pereira 21'), Pogba, Mata (Lingard 24 - Sanchez 43'); Rashford, Lukaku
LIVERPOOL (4-3-3): Alisson; Milner, Matip, Van Dijk, Robertson; Fabinho, Henderson (Shaqiri 72'), Wijnaldum; Salah (Origi 79'), Mane, Firmino (Sturridge 31')