The Reds were on top for long periods and peppered the United goal with 19 shots, but a combination of outstanding goalkeeping by David de Gea and poor finishing denied them a decisive moment, BBC reported.
The most noteworthy incident of an uneventful first half was De Gea denying Joel Matip from close range with a brilliant reflex save with his outstretched left boot, before Mohamed Salah fired the rebound wide.
United offered very little quality in attack although Simon Mignolet had to smartly parry Romelu Lukaku's fierce attempt before the interval.
Liverpool dominated even more after halftime but Emre Can wasted the best chance when he volleyed over the bar from eight yards after being picked out by the impressive Joe Gomez's precise cross.
This was only Liverpool's third clean sheet in 13 games this season but Mourinho will have been the happier manager at the final whistle.
The visitor had just 38 percent of possession – a 3 percent improvement on their last visit – yet United produced a dogged defensive display to stop Liverpool from scoring for only the second time this season in the Premier League.
‘Not tolerated at Liverpool’
Jurgen Klopp said after the game the defensive approach Manchester United adopted at Anfield would perhaps not be tolerated at Liverpool.
When asked by Sky Sports whether he was frustrated with Jose Mourinho's setup, Klopp said, "It's not my right to be frustrated.
"I'm sure if we would play like this then we could not do it at Liverpool, but obviously in Manchester it's OK. I don't judge this, it's all OK.
"We have to think about our performance, our situation and it's not worse. Yes, we wanted to win, 100 percent. We were working for it. All the subs were quite offensive. There were a lot of brilliant individual performances in a really difficult game," the German added.
Meanwhile, Mourinho questioned why Klopp had not shown the adventure to replace one of his three central midfield players with another attacker despite United fielding just two in Nemanja Matic and Ander Herrera.
Mourinho told Sky Sports, "In the first half, while we had energy in our midfield, we were very much in control. We had no problems and they had one chance. Apart from that we had total control.
"In the second half, I was feeling that I need my bench and I had no bench. I need to play against Henderson, Can and Wijnaldum with power and energy and no power, no energy.
"We were playing with only two midfield players and nobody else to compensate that. So progressively we were losing some control and they were having a little bit more initiative and in the end, a point.
"They kept three midfield players in the center of the pitch and I had no chance to answer."
Mourinho is likely to face more criticism for the way he approached the trip to Anfield but the United manager insisted it was a "positive" result.
It maintains his team’s unbeaten start of the season with games against Tottenham and Chelsea to come in the next three weeks.
Mourinho added, "It is easy for the experts to speak but it is more difficult when you are on this side. Liverpool midfield was really strong and they never changed for 90 minutes.
"I had no chance to bring the game in another relation so I think it's a positive point for us. We controlled them and it was quite frustrating for them."