Liverpool departed Old Trafford on Sunday with a 1-1 draw, but their hosts left it late to secure a point, as Zlatan Ibrahimovic headed in his 10th Premier League goal from his last 10 appearances six minutes from time.
James Milner had put Liverpool ahead just before the half-hour mark, converting a penalty after Paul Pogba handled in the area, but Mourinho is adamant that his men deserved more.
"It was entertaining, I have to agree, but it was not high quality," he told Sky Sports. "Both teams can do much better than we did in terms of quality.
"But it was entertaining because we created chances to be in front. They defended well and it was good to see they can play well defensively.
"We changed our way of play because Liverpool pressed very well. We went for a more direct approach and caused problems.
"We were the team who attacked, Liverpool were the ones who defended. I think we deserve more, but I think the point is something we have to accept.
"They stopped our consecutive run of wins, but they didn't end our run without defeat."
United's final substitution saw Mourinho introduce Marouane Fellaini ahead of Marcus Rashford and the Belgian played a key role in the equaliser.
Mourinho explained that his decision to put Fellaini on instead of the 19-year-old attacker came down to Liverpool's swelled numbers at the back and his desire to change to a more route-one approach.
"The people need to know what Marouane Fellaini is great at and what he is not so good at," he said to BBC Radio 5 live. "Marouane is very good in some aspects.
"I was looking to the pitch and only saw yellow shirts in the box. I thought it's difficult to penetrate so I thought we should go direct.
"If we had a few more minutes I think we would have done it [win]."