"Alisson doesn't have COVID. He told us after the [Brighton & Hove Albion] game [on Saturday] he felt it after 60-70 minutes, his hamstring. He's had a scan, a little one, but enough for today and probably another week," Klopp said.
Kelleher, 22, put in a solid performance on his Champions League debut, saving well from Noussair Mazraoui in the first-half and doing well from an Antony shot after the break, which saw David Neres strike the post with the rebound.
But his star moment came when he denied Klaas-Jan Huntelaar from point-blank range in the final exchanges to seal the 1-0 win.
"It was an amazing one for me, to make my debut in a Champions League game was massive and we got the win and qualified. A great night," said the Irish keeper.
"That's what I'm there for, I work every day in training to make sure I am ready and I was able to make a good few saves."
Liverpool have suffered a rash of injuries throughout this season, with players like Virgil van Dijk, Thiago Alcantara, and Trent Alexander-Arnold all missing significant time. Several other players have missed time due to testing positive to the coronavirus.
Klopp has been repeatedly vocal over player injuries, and even blamed British broadcasters Sky and BT over match scheduling that he says is leading to more absences.
"Why we change late is because we constantly have to think that somebody will go down with an injury. We cannot change early, because we change early and the other one has an injury, and you end the game with nine players," Klopp previously said after a match against Leicester City last month.
"It's about all the players, it's about the England players, it's about all the players who will play the European Championship next summer -- if you [Sky] don't start talking to BT, we're done."
When asked by BT Sport whether James Milner suffered a hamstring injury following Saturday's draw with Brighton, Klopp said: "Yeah, congratulations. Hamstring. Surprise. And they had injuries as well because it's a tough time."
The COVID-19 pandemic caused the previous season to finish late and the current campaign to be condensed into a shorter period.