An own goal from Jonny Evans and headers from Diogo Jota, continuing his streak of scoring in his first four league games at Anfield, and Roberto Firmino were enough to give Juergen Klopp a victory over his predecessor Brendan Rodgers.
Although the Reds were without Mohamed Salah, who had tested positive for COVID-19, and the injured Jordan Henderson and Trent Alexander-Arnold, as well as a string of other absentees, it was a strong showing from Liverpool who look ready to launch another serious title bid.
Leicester keeper Kasper Schmeichel pulled off a series of saves to deny the home side after the break while Firmino twice hit the post and had an effort cleared off the line.
The Brazilian, who had scored just twice in his last 23 Premier League games, got his reward four minutes from the end with a fine header from another Milner corner.
Klopp was full of praise for the way his team had won so convincingly against opponents who would have gone top with victory.
"We deserved it. The boys played an incredible game against a top, top, top opponent. I don't take things like this for granted. The boys were on fire football-wise," said the German.
"I liked the overall performance - the result and clean sheet. We defended incredibly well," he added.
Rodgers was unhappy with the way his team had approached the first half.
"We, first half in particular, were too passive and... gave away poor goals. We had a good opportunity on the counter with Harvey Barnes, but overall when you are passive like that your football is not quite the same, we weren't aggressive enough," he said.