Second-placed Liverpool have now netted 39 times in 13 matches and no player has more Premier League goals individually this season than any of the starting front three of Mohamed Salah (11), Sadio Mane and Diogo Jota (both seven).
Jota added a couple to his tally from a combined distance of seven yards, midfielder Thiago Alcantara made it two in two games shortly before half-time and Virgil Van Dijk killed off any fanciful hopes of a Southampton comeback just after the break.
It was the perfect preparation heading into a midweek Merseyside derby at Goodison Park and a reunion with former Liverpool manager Rafael Benitez.
The trend for Southampton, however, remains a downward one after last week’s defeat at Norwich.
Only three league wins all season - only one of those away from home - leaves them just five points above the bottom three with the four teams immediately below them all with a match in hand.
They had not won on this ground since 2013 and not scored on their last five visits so the omens were not good, but conceding the fastest goal of the Premier League season after just 97 seconds was the worst possible start.
The returning Andy Robertson - one of six changes from the midweek Champions League win over Porto - resumed his marauding left-back role after not starting the last two matches because of injury and made an immediate impact.
He combined with Mane down the left to get to the byline and pull back a cross for Jota - who seconds earlier had squandered a chance on the breakaway by delaying too long - to score from close range.
Mane’s header from Robertson’s free-kick was ruled out for offside while Salah curled a shot wide of the far post after robbing Mohammed Salisu on the edge of the area.
After Jordan Henderson ballooned over a good chance, Jota doubled his tally in the 32nd minute with the 700th goal of Klopp’s tenure after Southampton were cut open down their right side this time.
Thiago’s ambitious crossfield pass was recovered by Trent Alexander-Arnold, feeding Salah to then collect a clever return pass from Henderson and slide a low cross which Jota who could not miss from a couple of yards out.
The Portugal international was denied a first-half hat-trick by a block from Lyanco but the Southampton defender could only deflect the ball into the net when Thiago blasted a shot from just inside the area moments later after a delicious drop of the shoulder opened up the space
But, for all their superiority in the final third, Klopp’s side were still leaving the back door open and had the visitors been sharper they would have had something to show for their first-half efforts.
Armando Broja was denied by Alisson Becker after nipping in front of, and outpacing, Ibrahima Konate while Liverpool’s goalkeeper saved well down to his right to deny the unmarked Adam Armstrong, who looked certain to score.
Mane’s visible annoyance when Salah failed to find neither him nor Jota in a three-on-two break indicated there would be no letting up, and shortly after half-time the fourth goal duly arrived courtesy of a low Van Dijk volley from Alexander-Arnold’s corner.
Konate’s sliding block and recovery to deny Armstrong, together with Alisson, from a Saints counter-attack maintained a third successive clean sheet.
Additional reporting from PA