Kylian Mbappe nearly helped salvage a point for his side but Liverpool's talismanic Brazilian ensured Klopp's side picked up a win in their first game of the 2018/19 Champions League campaign.
Chelsea, third on 15, have the chance on Sunday to join Liverpool with a flawless start to the campaign on 18 points if they win at West Ham United while Watford's 1-1 draw at Fulham in the early game kept them fourth on 13.
It is the first time in Liverpool's history that they have begun a campaign with seven successive wins in all competitions, following Tuesday's late Champions League triumph over Paris Saint-Germain.
But the German praised Shaqiri's impact at the club and said he would eventually get his chance.
21' - CHANCE! Salah plays a lovely one-two with Firmino and looks poised to tuck home from close range when Cedric comes in from nowhere to make a brilliant last-ditch tackle.
The African forward was criticised for being wasteful in possession during Liverpool FC's 3-2 win over PSG in the Champions League.
Just before the break, after a towering header from Joel Matip off Alexander-Arnold's corner had doubled Liverpool's advantage, he stepped into the spotlight again. The Reds are cruising now.
"We were unfortunate with the opening goal, a ricochet off a couple of players and it ends up in the back of the net". Subs: Milner 6, Gomez 7, Keita 6.
Southampton didn't have a shot on target until the 91st minute.
The last Southampton player to score a Premier League goal against Liverpool is Sadio Mane.
Meghan Markle brings mom Doria Ragland to her Grenfell cookbook launch
The dishes include coconut chicken curry, aubergine masala, caramelized plum upside-down cake and spiced mint tea. After marrying Prince Harry on May 19, the former Suits star officially became a member of the royal family .