Klopp started with a 4-2-3-1 featuring Shaqiri in a number 10 role and the switch worked well, as the Swiss star's cross resulted in an own goal by Wesley Hoedt, before Mo Salah tapped in the third after his free-kick bounced down off the crossbar.
Shaqiri scored eight goals in 36 league appearances for Stoke City last season and was on target for Switzerland in their Nations League victory over Iceland earlier in the month.
Liverpool meanwhile had plenty of chances to increase their lead - sub James Milner had two very good shots in quick succession before Salah thought he had his second of the afternoon, only to be flagged for offside.
Despite the 3-0 victory over Southampton, Klopp has admitted Liverpool struggled due to a change in tactics. It's just good to have him.
Klopp had dismissed talk of a Salah goalscoring crisis with the Egypt global having drawn a blank in his last three games and the majority of Anfield was willing him to score every time he got in the penalty area.
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.
"I still need some time but I am on a good way and we have a lot of games, a lot of players, so I hope that I can also play a few games more".
"I lost seven games in a row once, that felt different!"
"They've got a really healthy group of players - they're the real deal".
'All the different challenges with the different styles of the opponent, the boys were there. "It is better to win these games than draw them or lose them, but I am really sure we are mature enough to understand it in the right way". It was a nice feeling to be on the field 90 minutes, ' added Matip, who underwent a thigh operation in April.