The 18-year-old was carried off on a stretcher and taken to hospital after he was on the receiving end of a challenge from Leeds defender Pascal Struijk who was subsequently sent-off.
And Klopp says while the club will carry on in Elliot's absence, they will also wait for him to return.
The German said: "We'll play without him, but we will wait for him as well because he is a top player."
In a statement that evening Liverpool confirmed the hugely talented youngster, who was signed from Fulham in 2019, had gone to Leeds General Infirmary.
They said: "Harvey, his family and Liverpool FC would like to place on record their thanks to the paramedics and all the staff at Leeds General Infirmary for the care and treatment given."
And Elliot himself took to social media afterwards and thanked people for their messages of support.
He said: "I'm of course absolutely devastated with what happened yesterday at Leeds, but have been totally overwhelmed by the love and support shown to me by the entire football world following the injury.
"Thank you so much to everyone who has reached out or sent messages to me and my family, it means so much to us.
"Also a massive thanks to everyone inside Elland Road for the reception you gave me immediately after it happened.
"I am now fully focused on my recovery and will give everything in my rehabilitation to be back out there as soon as I can."