The game was halted shortly before the hour when Elliott, who has become a member of Liverpool's first-team this season, was hurt in a challenge by Leeds player Pascal Struijk.
Jurgen Klopp said after the game that he had been told Elliott's ankle had been dislocated.
"Harvey Elliott is in hospital, a bad ankle injury for sure. It looked like it was dislocated, the medical department put it back. Massive pain, shock for him, for us and we don't know more. We have to wait," he said.
"I couldn't care less [if it was a red card], it is not my business. It is a serious injury, definitely for an 18-year-old boy. The red card is not important. Two or three weeks we can play on, Harvey will not.
"We played a really good game until Harvey had to go, the whole team was shocked and we lost rhythm. That is normal, human. We played as good as we can against Leeds.
"I spoke to the boys afterwards, we have to speak a little bit about the football but Harvey overshadows it."
The club confirmed in a statement later on Sunday that he will udnergo surgery.
"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," the statement read.
"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."
The 18-year-old was about to be substituted for Jordan Henderson when he was injured.
The Liverpool physios were on the pitch before referee Craig Pawson could stop play, while TV pictures showed Mohamed Salah visibly upset by the injury and Klopp remonstrating about the severity of the challenge.
Pawson sent off Struijk for the incident shortly before Elliott departed on a stretcher.
Sources told ESPN that Pawson spoke to his fourth official Andy Madley, who was close to the incident, before deciding to dismiss Struijk for the intensity of the challenge and endangering the safety of an opponent.
Elliott posted on Instagram: "Thank you for the messages guys. Road to recovery. YNWA [You'll Never Walk Alone]."
Leeds boss Marcelo Bielsa said postmatch: "I regret dearly what happened, I wish it did not happen. For that to happen to any player is saddening. For a young player playing at a high level even more.
"I am sure my player did not have the intention to harm him in any way. Perhaps what happened after he recovered the ball caused the injury. The tackle was clear, but clearly the acceleration at how our player moved caused it. In no way did he have provoke any harm to the opponent."
Leeds players also extended their support to Elliott.
"First and foremost our thoughts go to Harvey Elliott. It's a horrible injury and we hope he gets back on the pitch as soon as possible," Stuart Dallas said.
Elliott joined Liverpool from Fulham in 2019 and impressed on loan at Blackburn Rovers last season.