As if to cement their controlling performance against the league's second-place team, Klopp's side restricted the Premier League's leading scorer, Jamie Vardy, to one off-target shot in his worst performance of the season.
Alexander-Arnold supplied assists for both of Roberto Firmino's goals before completing the scoring with a stylish breakaway strike.
Mikel Arteta got off to an underwhelming start as Arsenal manager with a 1-1 draw at Bournemouth, but Carlo Ancelotti had a more positive impact on his Everton bow as Dominic Calvert-Lewin's brilliant diving header earned a 1-0 win over Burnley. Good to get the goals the three points is what we came for. But we are not even halfway [through the campaign].
He went on to praise their performance on an evening which saw Trent Alexander-Arnold run the show.
"We played our last two games against two of arguably the best teams in the history of the Premier League", Rodgers told Match of the Day.
The Leicester goalkeeper did well soon after the first Liverpool goal when he got a hand to a Sadio Mane hit from close range.
Klopp's males lastly had the respiratory area they desired through the arm of Caglar Soyuncu as he dealt with Alexander-Arnold's nook 20 minutes from time. They left the whole right side open for us.
If Leicester had any hope of repeating its remarkable 5,000-1 title triumph from 2016, it surely ended with this first home loss of the season. Leicester is now just one point ahead of third-place Manchester City.
Their last Premier League defeat came 358 days ago when they lost to Manchester City. Michael Oliver made the right decision and VAR confirmed it.
Another defensive error then teed up Martial to dink over Dubravka for the fourth early in the second half.
Leicester's nervousness in defence was evident in the fifth minute when Jonny Evans gave the ball away just outside his own area to Georginio Wijnaldum who, along with Firmino, teed up Salah, but the Egyptian could only spoon his effort over the crossbar from 18 yards.
It would seem that the only people who don't yet believe that the Premier League trophy is on its way to Liverpool are the very supporters who are so desperate to witness it.
Liverpool: They won't be well rested after playing an extra 30 minutes on Saturday and then having to travel back from Qatar, so it will be interesting to see if Jurgen Klopp rests some of his players.