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 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.
Job done! ✅@LFC battled hard against a resilient @Wolves side to seal a 1-0 win and keep up their #EmiratesFACup… https://t.co/OSqSv3Kxlp
— Emirates FA Cup (@EmiratesFACup) 1673993134000
“It feels like ages ago that we had that feeling that we win and play well,” Klopp said.
“We played a really good first half and in the end it was just passion to block the shots. It’s the reaction we wanted to see from ourselves and the boys delivered.
“The goal was exceptional. I don’t want to make Harvey’s performance smaller, but I had a few men of the match today, just the others didn’t score.”
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 claimed 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.
Sa was caught too far off his line but that should not detract from the quality of Elliott’s first goal since October.
Snapping into tackles and winning second balls, Liverpool were showing more hunger and tenacity than in recent weeks.
Gakpo was still searching for his first Liverpool goal in his third appearance since signing from PSV Eindhoven.
The Netherlands forward nearly broke his duck with a dipping volley that flashed just over.
Gakpo was replaced by Salah for the last 25 minutes, but it was Wolves who finished strongly with Ruben Neves’ free-kick narrowly clearing the crossbar.
Wolves’ best chance to draw level came when Jimenez couldn’t head home on the goalline from Traore’s cross as Joe Gomez got the crucial block.
Elsewhere, Championship side Birmingham survived a scare in their 2-1 win at third-tier Forest Green.
In a delayed third-round clash after the original tie was postponed due to a waterlogged pitch, Ben Stevenson’s brilliant long-range strike put Forest Green ahead after eight minutes.
But Lukas Jutkiewicz grabbed Birmingham’s equaliser in the 50th minute and Kevin Long killed off League One’s bottom club 15 minutes later.
Bristol City won 2-1 at Swansea, while Luton beat Wigan 2-1 and West Brom thrashed Chesterfield 4-0.