The hosts’ midfielder Pascal Struijk was sent-off for a challenge that appeared to break the 18-year-old’s left leg midway through the second-half.
By that point Liverpool were already comfortably ahead thanks to Mohamed Salah‘s 100th Premier League goal and Fabinho’s 50th minute strike.
Sadio Mane rounded off the scoring in stoppage time.
However, the loss of Elliott cast a dark cloud over a positive performance for Jurgen Klopp.
Elliott, who made his Premier League debut as a 16-year-old for Fulham, had broken into the Reds side with impressive early season displays against Burnley and Chelsea.
Liverpool’s title defence was blown off course last season by a series of long-term injuries to key players such as Virgil van Dijk, Joel Matip, Joe Gomez and Jordan Henderson.
Van Dijk’s return from cruciate knee ligament damage has made a massive impact with Liverpool conceding just once in their opening four games.
However, Salah aside, there remain questions over whether Liverpool have the firepower at the other end to match their rivals for the title in Chelsea, United and defending champions Manchester City.
The visitors should have been out of sight before half-time as they quietened the Elland Road crowd, enjoying just their second sight of a Premier League games in 17 years.
Salah rounded off a brilliant team move to open the scoring and bring up his century in just 162 games in the English top-flight as he and Matip exchanged a one-two before Trent Alexander-Arnold’s cross picked out the Egyptian for a tap in.
Mane’s form remains a concern for Klopp despite his late strike as the Senegalese missed a series of golden chances, including when he spooned over an open goal from Diogo Jota’s cross.
After Mane then failed to convert a one-on-one with Meslier minutes into the second-half, Liverpool finally doubled their lead from the resulting corner.
Fabinho swept home a loose ball after Van Dijk failed to connect cleanly with a free header from a corner.
Unfortunately, the major talking point was still to come as Struijk won the ball in a sliding challenge but the momentum of the Dutchman’s trailing leg took him through the back of Elliott.
The incident took place right in front of the Liverpool bench and Klopp was involved in an angry exchange of words with Leeds fans, who had initially been angered that the game was stopped.
Naby Keita, Curtis Jones and Ibrahima Konate were among the Liverpool substitutes to cover their eyes in shock at the injury.
However, Elliott was able to applaud the Leeds support as he was carried from the field as they gave the England under-18 international a rousing reception.
Rather than make the man advantage count, Liverpool were slack in the final quarter and could have paid when Patrick Bamford’s effort from nearly halfway had to be tipped over by Alisson Becker.
Mane also missed two more glorious chances before finally scoring his second goal of the season when he span onto Thiago Alcantara’s pass and blasted into the car corner.