Rangnick was without several regulars, including Cristiano Ronaldo, mourning the death of his newborn son
Mohamed Salah scored twice as Liverpool thumped old rivals Manchester United 4-0 at Anfield on Tuesday to move top of the Premier League, for 24 hours at least.
After a 5-0 win at Old Trafford in October, this was another humiliation of United by Juergen Klopp's team, who were barely threatened by their lifeless opponents.
Liverpool moved to 76 points from 32 games with title rivals Manchester City on 74 points from one game fewer, with Pep Guardiola's side hosting Brighton and Hove Albion at the Etihad on Wednesday.
It was a miserable night for United who are now without a win in their last eight league clashes with Liverpool and have failed to score in five of their last six trips to Anfield.
"It is embarrassing, it is disappointing, maybe even humiliating, we have to accept that they are six years ahead of us now," said United's interim manager Ralf Rangnick who added the club now faced a "rebuild".
United dropped below Arsenal on goal difference, falling to sixth place on 54 points, having played two games more than the Gunners.
Tottenham Hotspur currently occupy fourth spot on 57 points but even though United remain in with a chance of qualifying for the Champions League there was nothing in their performance that indicated they merited such a reward.
Rangnick, who was without several regulars including Cristiano Ronaldo, mourning the death of his newborn son, opted to play five at the back with Phil Jones drafted in.
IN CONTROL
It was an approach that simply did not work and Rangnick abandoned it at halftime but by then Liverpool were 2-0 up and in control.
Colombian Luis Diaz gave Liverpool a fifth-minute lead, tapping in from close range after Sadio Mane set Mohamed Salah free down the right and the Egyptian delivered a pinpoint low cross for a simple finish.
United's problems got worse in the 10th minute when Paul Pogba limped off injured and with Rangnick's side struggling to get any foothold in the game, it was no surprise when the home side doubled their advantage.
Mane again provided the spark with a superb first-time pass finding Salah who, having lost his bewildered marker Diogo Dalot, cut in from the right and provided his trademark left-foot finish.
The Senegalese creator turned finisher in the 68th minute, making it 3-0 after Andy Robertson broke down the left and then Diaz picked out Mane whose first time finish on the turn flew into the bottom corner.
Salah completed the rout in the 85th minute as substitute Diogo Jota threaded a pass through to the Egyptian, who beat David De Gea with a deflected finish that came off the leg of the sliding Aaron Wan-Bissaka.
The Anfield crowd revelled in the humiliation of their North West rivals while a section of the visiting fans chose to leave early.
Klopp gave the Kop four celebratory fist pumps after the whistle but when asked about the 9-0 'aggregate score' against United this season, showed some sympathy for his compatriot Rangnick.
"It is not a normal situation. They are not in a good moment and on top have a lot of injuries. When Pogba left the pitch they played without their usual midfield. The pitch can become really big with a lot of offensive players on it.
"Centre midfield with (Nemanja) Matic and (Bruno) Fernandes is not how you want to play. Nothing against the players, it is just not their natural game. We had 70-75 per cent of the ball and they have to defend and that is not easy. I feel a bit for them."