Klopp made nine changes to the side that lost narrowly for the first time in the Premier League this season to Manchester City on Thursday, with Lovren one of only two players to retain their place.
However, the Croatian pulled up with a hamstring injury after just six minutes and was replaced by 16-year-old Ki-Jana Hoever.
"Hamstring, is what I heard — without any signs before," said Klopp on what forced Lovren off. "I asked everybody, no signs, nothing, just out of the blue, so that's the decision you have to make. I am not sure what you all would have said if immediately from the beginning if our centre-half situation was Fabinho and Ki-Jana; then probably a few very smart people would tell me that I don't respect the competition or whatever."
Goals from Raul Jimenez and Ruben Neves either side of half-time sent the hosts into round four, where they will face Shrewsbury or Stoke City, despite Divock Origi's fine strike briefly bringing Liverpool level after the break.
Hoever's introduction meant Liverpool fielded three teenagers for the majority of the match with Rafael Camacho and Curtis Jones also handed debuts.
However it was the visitors' most experienced player, Milner, who was at fault for the opening goal when Diogo Jota caught the former England international in possession in midfield and put Jimenez clean through for the Mexican to score his seventh goal of the season.
Without Mohamed Salah, Roberto Firmino and Sadio Mane, Liverpool were struggling to get any momentum going forward.
But the visitors were level out of nothing when Origi collected a loose ball on the edge of the area six minutes into the second period and smashed the ball high past John Ruddy.
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