Karius carried his poor form into pre-season, with Liverpool fans calling out for a replacement after mistakes in friendlies against Tranmere and Borussia Dortmund.
The Liverpool captain only returned to training on Sunday, but Klopp did not rule him out of contention to feature against West Ham on the weekend.
Liverpool were in complete control from the start and were awarded a penalty in the 16th minute. Sadio Mane stayed on his feet to get a shot away but referee Michael Oliver called play back for a foul on the Senegalese winger by defender Bremner.
The Brazilian's miss did not haunt his Anfield bow for long as compatriot Firmino put the Reds ahead soon after.
Three minutes later, Wijnaldum rounded off a well-worked team move with a neat finish to double the hosts' lead.
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Some last-ditch defending prevented Liverpool from regaining their two-goal cushion on the stroke of half time, with Sirigu thwarting Wijnaldum after the midfielder had made another run beyond the Liverpool attack, and Alex Berenguer cleared Alexander-Arnold's follow-up off the line.
It was a moment which pleased Klopp as much as the performance.
Alisson was making his first appearance for the Reds in Dublin following his World Cup campaign with Brazil and according to Scotland defeder Roberston, the new boy has settled in well on Merseyside.