In a mediocre encounter, United stopped the rot that had been spreading after the heavy home loss to Bournemouth and a tame Champions League exit rubber-stamped by Bayern Munich at Old Trafford.
While this was hardly a classic display from injury-hit United, who had Diogo Dalot sent off for dissent in stoppage time, Ten Hag had cause to be much happier than Liverpool's Jurgen Klopp whose side missed a chance to return to the top of the Premier League.
United keeper Andre Onana made one fine first-half save from Virgil van Dijk, while Rasmus Hojlund had the visitors' best chance in the second half but saw his shot blocked at the near post by Liverpool's Alisson.
Liverpool dominated possession but were too often wasteful and had to settle for a point, not enough to overhaul Arsenal whose earlier win against Brighton took them to the summit.
'Precious point will calm the mood for Man Utd'
Liverpool falter on Anfield's big day
This was a red-letter day in the history of Liverpool's famous old stadium as the new top deck of the Anfield Road Stand was in use for the first time.
It made the attendance of 57,158 the biggest for any Liverpool home game since the FA Cup tie against Burnley in February 1963 and the third largest at home in their league history.
This was not a game to match the landmark.
Nine months after Liverpool swamped United in a 7-0 Anfield rout, the Red Devils never looked like facing a repeat of such carnage.
Liverpool went into this encounter with their arch-rivals as overwhelming favourites after their rise to the top of the table and United's own struggles, but the reaction at the final whistle made it clear this was two points lost for Klopp's side rather than one gained.
The hosts came out at lightning speed, clearly determined to shatter United's fragile confidence, but they lost momentum as this became an undistinguished, scrappy game, almost devoid of real quality.
Liverpool's 69% of possession was reflected in 34 shots, but too much of their work was wayward, laboured and kept out by a resilient United defensive performance.
And there was no late show to snatch victory, which has been a Liverpool trademark in a season that has brought victories but only intermittently seen Klopp's men touch the heights.
Even Mohamed Salah was unable to produce a piece of magic and when the big moment came late on, substitute Cody Gakpo headed over when unmarked.
United show battling spirit
The odds were heavily stacked against Ten Hag's United as they arrived at Anfield in the wake of heavy recent criticism, to be tasked with facing a Liverpool team who started the day at the top of the Premier League and who also handed out that historic thrashing here last season.
United had to survive a whirlwind Liverpool start and while they were short on creativity and grateful for the home side's disappointing work in front of goal, the visitors deserved credit for digging deep with a defensive display of real vigour.
Harry Maguire was missing after being injured against Bayern, yet there was much to be admired about how United were willing to throw bodies in front of everything to halt Liverpool, and keeper Onana dealt with everything that came his way.
While United could be satisfied with a point in the circumstances, they will also know striker Hojlund had a big opportunity in front of the Kop in the second half, only to shoot straight at Alisson.
Given the circumstances, however, United will take this result.
