Elliott was introduced off the bench alongside new signing Luis Diaz after Diogo Jota’s header had opened the scoring early in the second half.
Diaz made an instant impact by teeing up Takumi Minamino to double Liverpool’s lead.
But the biggest cheer was reserved for Elliott when he fired home on the volley 14 minutes from time.
Rubin Colwill pulled a goal back for the Bluebirds, but it is Liverpool who will meet Norwich at Anfield in the last 16.
For the final time, Jurgen Klopp was without Sadio Mane and Mohamed Salah due to their participation at the Africa Cup of Nations with the club teammates set to face off in Sunday’s final as Senegal take on Egypt.
After two weeks without a game, Klopp took the opportunity to name a strong side.
But the home side were frustrated for the first 45 minutes as Dillon Phillips produced a brilliant save to deny Jota and Minamino had a goal ruled out.
Jota finally made the breakthrough with a fine header from Trent Alexander-Arnold’s free-kick.
Both Diaz and Elliott were then given a huge ovation as they made their entrance.
Liverpool beat Tottenham to the signing of the Colombian for an initial £37.5 million ($50 million) and Diaz showed his quick feet to pounce on a Cardiff defensive error and pull the ball back for Minamino to finish.
There were fears Elliott could be out for at least the rest of the season when he suffered a dislocated ankle against Leeds.
The 18-year-old is one of the brightest young talents in English football as he demonstrated with a sublime touch and finish to smash home from Andy Robertson’s cross.
A much-changed Cardiff at least got a consolation to remember from their trip to Anfield 10 minutes from time when teenager Colwill rounded off an incisive counter-attack.