The Reds, without a number of key players, took a surprise first-half lead against the run of play when James Milner scored from the spot after Paul Pogba's handball inside the area.
They had chances to build on their advantage beyond the break, although Roberto Firmino couldn't capitalise upon their best, but it looked as though they were set for a crucial three points to return to second in the league.
But Zlatan Ibrahimovic's stooping header, coming after United substitute Marouane Fellaini had clipped the post with a header, in the 84th minute denied the visitors a morale-boosting first win of 2017.
Klopp: Over the course of the entire game, we would have deserved the win
In his post-match press conference, Klopp insisted that he was "fine with the performance", adding: "When you see the line-ups and see the circumstances, you would say 'A point at Old Trafford, come on, take it, go home and don't think about it. Well done.' Unfortunately, it doesn't feel exactly like this and that's because of the performance of my boys."
The German believes Liverpool "were the better side" over the course of the 90 minutes and said that it was "difficult" for his side to deal with the height of physicality of Fellaini "after 75 minutes of dominating the game" and "then having all the balls bouncing around the 18-yard box."
He added that Liverpool "made this mistake", acknowledging that he had "no idea how it exactly happened" but that they "could have cleared the situation already" and then "it comes back from the post" for Ibrahimovic's leveller.
"They were then more awake and used the situation," Klopp said, insisting that they had "two unbelievable chances" before the equaliser with Firmino's "big chance" shortly after Philippe Coutinho's introduction.
The manager added that he can "easily accept" the draw and declared that the result wasn't "a problem" although, "over the whole 98 minutes or whatever" he believed they "would have deserved the win."
He continued: "That's absolutely not interesting for anybody but it's important for me to know about the attitude and character of my boys, because they played with a lot of little problems."
Klopp said that they "don't make big things of it" but came into the game with "a few issues, smaller or bigger" and praised his players for doing "really well" and insisting he was "happy."
Reds boss praises "brilliant" Alexander-Arnold
Klopp handed 18-year-old Trent Alexander-Arnold his full Premier League debut in place of the injured Nathaniel Clyne, who missed out with a rib issue.
The West Derby-born teenager had only one minute of top-flight experience prior to his run-out against United after coming on as a late substitute in last month's victory at Middlesbrough.
However, Klopp was delighted with the Academy prospect's performance, insisting: "Nice, eh?"
He admitted that Alexander-Arnold "needed a little bit of time to adapt" after "the first two situations it was difficult" against Anthony Martial, a player he called "one of the quickest in the Premier League."
"But then he did brilliantly. All the boys like him and so they wanted to help him," Klopp continued, admitting that the decision to start Alexander-Arnold "was not that easy" because his "first impulse" was to play "Millie [James Milner] on the right and Alberto [Moreno] left."
The Reds boss went on to say that he avoided that because he "didn't want to change two positions because of one injury" and acknowledged that "with a new system" to "change both full-backs" he "didn't feel too well" about.
Yet he added that the club "didn't have to be too brave" to start Alexander-Arnold, because they "thought he was ready for it", adding that his performance "proved" that.
Klopp was also to asked to reflect on the performance of midfielder Emre Can, who bounced back from a poor display in the mid-week defeat to Southampton in the EFL Cup semi-finals.
He insisted that the German "did well, absolutely" but insisted that is what he expects from him, adding: "This is his quality. You cannot be surprised when he has a good game."
Klopp also warned that you "should not be surprised" when Can "sometimes has an average game" because he's "still a young player" but declared that his display against United represents "his level" and "what he has to show" regularly.
There will be a moment when we are the better side, insists Klopp
Despite the merit of a point away at Old Trafford, with a weakened team against a United side that had won nine straight games and gone 15 unbeaten in all competitions, the result means Liverpool fall further behind Chelsea.
The league leaders, who travel to Anfield later this month, are now seven points ahead of the Reds - who slip to third on goal difference after Arsenal's victory the day before.
But Klopp said that he "didn't see" the matches of Liverpool's rial teams, insisting that they "cannot look at it" and that his team are "in the region of the table where we have to be."
"In this moment, maybe for us it's a little bit more difficult to play our best because not everybody is available and all that stuff," he continued. "But everybody could see we will fight for it 100 per-cent."
Klopp declared that "there will be a moment" when Liverpool "are the better side" and when they have everybody "back, in shape, healthy, fit" and then they can "see how the base is then."
He said that given the circumstances, Liverpool's performance at United was "absolutely more than okay" because it is "not usual that a team" like that "do it in this situation after the game in mid-week and all that stuff", calling it "fine" and saying they must now "carry on."