Jurgen Klopp's side came in for scathing criticism after losing their last two Premier League games in desultory fashion at Brighton and Brentford.
But Elliott's brilliant long-range effort in the first half at Molineux secured a much-needed first victory in four games in all competitions.
Ironically, Liverpool's reward for seeing off the Wolves is a return trip to Brighton in the fourth round.
The Reds' dismal 3-0 defeat at Brighton on Saturday left them languishing in ninth place in the Premier League, 10 points adrift of the top four.
In the bout of soul-searching that followed that embarrassment, Klopp said his struggling team needed to go "back to basics".
Amid mounting questions about Klopp's own future at Anfield, the German said this week that he will not leave the club unless he is forced to and hinted at changes in his squad during the summer.
Hamstrung by injuries to a host of key players including Virgil van Dijk and Luis Diaz, Klopp made eight changes after the Brighton debacle, with only Cody Gakpo, Thiago Alcantara and Ibrahima Konate retaining their places.
Mohamed Salah and Alisson Becker were among those left out, but Liverpool's understudies rose to the challenge.
Wolves had forced the replay with a 2-2 draw at Anfield, but Julen Lopetegui's side were angered when they had a potential winning goal from Toti controversially ruled out for offside in the final seconds.
They were unable to take their chance to avenge that injustice despite a lively start that saw Adama Traore's cross headed just wide by Raul Jimenez.
It was Eliott who produced the opener in breathtaking fashion in the 13th minute.
Klopp's midfield has been written off as ageing and past their best this season.
But the 19-year-old Elliott offers hope for the future, a point he underlined by running unchecked from the halfway line before blasting a superb strike over Jose Sa from 25 yards.