United, boosted by David De Gea passing a fitness test, had initially two changes from the 1-0 defeat at Newcastle a fortnight ago as Aaron Wan-Bissaka replaced Diogo Dalot at right-back and Juan Mata made way for Victor Lindelof.
However, Marcos Rojo was later drafted into the side in place of Axel Tuanzebe after he suffered an injury in the warm-up.
Liverpool made three changes to the side that beat Leicester 2-1 at Anfield as the injured Mohamed Salah was replaced by Divock Origi, Joel Matip returned in defence in place of Dejan Lovren and Alisson was back in goal ahead of Adrian.
Liverpool went into the game looking to stretch their 100 per cent start to nine games and move level with Manchester City’s Premier League record of 18 consecutive wins.
But it was United who started positively without really threatening, while the first effort in anger came at the other end on 13 minutes when Georginio Wijnaldum’s low strike was comfortably saved by De Gea.
Scott McTominay stung the palms for Alisson with an effort from distance while De Gea was again called on to deny Roberto Firmino.
United took the lead on 36 minutes amid huge controversy as Lindelof appeared to foul Origi from behind before the ball was played to Daniel James whose cross from the right was superbly converted from close range by Marcus Rashford.
However, despite going to the Video Assistant Referee (VAR) the goal was allowed to stand much to the annoyance of Reds’ boss Jurgen Klopp.
VAR again came to the fore on 44 minutes when Mane got the better of Lindelof inside the area and then finished past De Gea.
However, replays showed the ball bounced off the Senegal international’s hand in the build-up and the goal was correctly chalked off.
United conceded ground to Liverpool after the interval, but goalmouth action was at a premium.
Substitute Alex Oxlade-Chamberlain had a brief sight of goal on 65 minutes but dragged his shot on the turn wide of the upright.
McTominay and Fred both tried their luck at the other end but neither managed to trouble Alisson between the sticks.
With Liverpool pushing more men forwards in search of an equaliser, United enjoyed a couple of breaks and Fred almost grabbed a killer second goal on 83 minutes, but his left-foot strike curled just wide of the far corner.
Liverpool finally broke through five minutes from time when Andrew Robertson played a cross in from the left and Lallana stole in at the back post to score his first goal in almost 18 months with a tap-in.
The visitors looked the more likely to win it in the closing stages as Trent Alexander-Arnold and Oxlade-Chamberlain went close.
However, United held on to claim a point which moves them up to 13th, while Liverpool’s lead at the top over Manchester City is now six points.