Elliott, 18, appeared to dislocate his left ankle in a second-half challenge with Leeds substitute Pascal Struijk and was carried off the field on a stretcher after lengthy treatment.
Struijk was shown a straight red card by referee Craig Pawson after a VAR review and the injury clearly distressed Elliott’s team-mate Mohamed Salah, while Liverpool’s medical staff immediately leapt to the player’s assistance.
Elliott posted a picture of himself with thumbs up on Instagram before full-time, writing: “Thank you for the messages guys! Road to recovery. YNWA.”
Klopp, whose side moved level on points with top two Manchester United and Chelsea after goals from Salah, Fabinho and Sadio Mane at Elland Road, said: “One of the Instagram posts I liked, if he did that already it is a sign he is not in much pain, so that is great.
“We all saw the ankle was not in the right place and the medical team put it in the right place.
“There is not a lot of good things to say, otherwise it is a bad injury. We have to wait for further examinations and scans.”
Elliott was taken to hospital in Leeds and, although discharged in the evening, Liverpool confirmed the injury was “serious” and the player would need an operation.
“Elliott will be required to undergo surgery in the coming days and the club will offer a further update on his prognosis in due course.
“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.”
Struijk wished Elliott well in a post on Instagram, writing: “In today’s game something happened that I would never wish on anyone.
@harveyelliott07 my thoughts are with you. I’m gutted and never meant for this to happen. I wish you a speedy recovery and hope to see you back on the pitch soon.”
Liverpool skipper Virgil van Dijk, who recently returned to action after being sidelined for 10 months due to a knee injury, told Sky Sports: “I have experienced, with all the players and staff, everyone around Liverpool and the fans – they had my back.
“They were there for me in difficult times. I am 100 per cent sure we are all going to be there for him no matter what.”
Elliott’s injury marred what was a convincing win for Klopp’s side, who could have won by a bigger margin after creating 30 attempts on goal.
Klopp added: “Obviously we played a really good game today. We pressed exceptionally well and that’s why we won a lot of balls.
“It was far from perfect and we should have had more clear-cut chances. But that’s not a problem, we won 3-0.”
Marcelo Bielsa insisted Struijk, a 33rd-minute replacement for Diego Llorente, had made a genuine attempt to win the ball, but would not be appealing against his defender’s dismissal.
Bielsa said: “Pascal recovered the ball. The action to get the ball didn’t have an infraction, but I have the feeling that the movement of the rest of his body after he recovered the ball made him hit Elliott.
“These types of recoveries of the ball are totally frequent. The difference here was Pascal’s speed and power hit the opponent and that caused the injury.
“I regret dearly that such a young player who was playing at a high level was interrupted through this. I hope he is able to get back.
“For me the referee decides. He has enough resources to make the decision and I don’t know why the decision wasn’t immediate and it was after. But I never like to condition the decisions the referee makes.”
Leeds had their moments in a pulsating encounter – Rodrigo, Luke Ayling and Patrick Bamford had chances – but lacked Liverpool’s potency in the final third.
Bielsa said he took responsibility for his side’s defeat, adding: “As the first half wore on we started to lose control of the ball, it was difficult for us to recover it and when we recovered it it was difficult for us to put passes together.
“And the way Liverpool performed allowed their creative players to shine and the way I proposed the game did the opposite.”