Liverpool move six points clear at the top of the table, with a game in hand to come, over second-placed Arsenal.
But this was an opportunity lost for Arne Slot's men after they came from behind to lead 2-1 through Cody Gakpo and Mohamed Salah's penalty.
"Of course it feels for us like two points dropped," said Slot.
United broke their six-year goal drought at Anfield to open the scoring in spectacular fashion through Lisandro Martinez and got the point a much-improved performance deserved when Amad Diallo levelled on 80 minutes.
A point edges United up to 13th in the table and seven clear of the relegation zone.
"I'm pretty upset. If we show this level at Liverpool, first in the league, why can't we do this everywhere? It frustrates me," said United captain Bruno Fernandes.
"Also, finally I did a proper performance. We said we need much more from ourselves to get more from this season."
Slot warned that United were "much better" than their shocking league position suggested ahead of the game and so it proved.
United had been carved open at will by Newcastle in a dismal 2-0 defeat on Monday that left manager Ruben Amorim declaring they were in a relegation battle.
Amorim's hand was strengthened by the return of Fernandes and Manuel Ugarte from suspension, while Kobbie Mainoo also returned to the starting line-up.
The Portuguese's plan to frustrate Liverpool worked but the home side did have the chances early on to make the breakthrough in the first half.
Gakpo prodded past the far post before Alexis Mac Allister's volley forced Andre Onana into a fine save with his feet.
United took time to begin to impose themselves as an attacking threat but should have been in front before the break.
Rasmus Hojlund then had the best chance of the first half when he raced in behind the Liverpool defence but could not beat Alisson Becker.
Martinez blasts home
However, confidence was suddenly coursing back into United players who have looked bereft in recent weeks.
The visitors' breakthrough came from an unlikely source as Martinez showed his strikers how to finish with a blistering hit in off the underside of the bar for just his second goal for the club.
The Anfield crowd were beginning to get restless as the league leaders were briefly at sixes and sevens.
However, Liverpool's position at the top of the table owes much to the depth of firepower they possess.
Even on a quiet day for Salah, they had an in-form forward to come up with the moment of magic required to turn the game.
Gakpo turned inside his Dutch international colleague Matthijs de Ligt and blasted high past Onana for his 10th goal in 15 games on the hour mark.
Mac Allister's header then flicked off De Ligt's outstretched arm inside the penalty area and a VAR review resulted in the awarding of a spot-kick.
Salah smashed low and hard beyond Onana for his 18th Premier League goal in 19 games.
However, United's spirited display got the point it merited to halt an alarming losing streak for Amorim so early in his reign.
Alejandro Garnacho's attitude has been questioned by his new boss but the Argentine came off the bench to make the equaliser as his low cross was turned in by Diallo.
And it should have been even better for United when Harry Maguire spooned Joshua Zirkzee's pass over the bar with the goal gaping deep into stoppage time.
