Fabinho has performed admirably in defence since stepping in for Virgil van Djik, who suffered an ACL injury in the Merseyside derby against Everton earlier this month.
The Reds struggled in the first half against the Danish champions but took the lead through Diogo Jota 10 minutes after the break.
Substitute Mohamed Salah sealed the victory with a stoppage-time penalty which he had won himself.
But the status of Fabinho was on Klopp's mind after the final whistle at Anfield.
"Exactly the last thing we needed," the Liverpool manager said to BT Sport. "I don't know, he felt his hamstring so that is not good.
"He didn't feel it that much - he said he could have played on but not sprinting, so that doesn't help.
"We will see, we will have to do a scan and see but clearly it is not good.
"It was a tough night. It is a bit like in a marriage, there are good and bad times.
"It is not a bad time but it is a tricky time, 100% we have to stick together and fight harder and that is what the boys did.
"We are doing really well but it is a shock for a team because now we have to sort that injury."
Klopp added at his post-match press conference: "We lose pretty much one of the last men standing at the back, you bring on a 19-year-old kid and no one knows how that is going to work out. But Rhys did really well, he was commanding.
"We have to wait after a scan obviously. You know in these times when someone has a muscle problem and it is pretty rare the doctor calls me the next morning and says it is a false alarm, all fine and he can go again.
"So now it is only about how long, that is all. Now a week means three games, 10 days means four, 14 days means seven games and all these kind of things, so that is how it is.
"Yes, it is absolutely not cool but that it is the situation."
After Saturday's home game against West Ham, the Premier League champions have a crucial trio of matches in November against Atalanta, Manchester City and Leicester City.
When asked if Joel Matip would be fit enough to return and partner Gomez at centre-back against West Ham, Klopp said he did not know.