The Reds ended City's unbeaten league run this season in a thrilling 4-3 victory on Sunday night (14 January) in what was their first league game since the big-money sale of Philippe Coutinho.
Goals from Alex Oxlade-Chamberlain, Roberto Firmino, Sadio Mane and Mohamed Salah were enough for the home side despite a late fightback from the visitors.
Karius notably started the game for just his fourth league appearance this season, having constantly rotated with Simon Mignolet since arriving at Anfield in the summer of 2016.
The 24-year-old was, however, seemingly at fault for Leroy Sane's first-half equaliser which crept through Karius from a tight angle.
Late consolation goals from Bernardo Silva and Ilkay Gundogan meant the German had also conceded three goals from the four shots City had on target on the night.
Nevertheless, Karius did enough to convince Klopp to make him the new number one, as the German manager spoke about the need for an established goalkeeper between the sticks.
"Yes, he is the new number one," Klopp told DAZN after the game via FourFourTwo.
"With goalkeepers you have to commit yourself, in other positions you can change, as you want, and we do now - Loris is now in it."
Klopp also addressed Karius' blunder for the Sane goal, citing that City's goalkeeper in Ederson, who has been impressive since joining Pep Guardiola's side in the summer, made a crucial mistake that led to Salah making it 4-1.
"Yes, of course, it's the corner where he was. You showed it to me," Klopp added. "People always find the hair in the soup. Loris knows it."
"The Man City goalkeeper probably caused Salah's goal and he is still a world-class goalkeeper. But nobody will talk about that."