Harvey Elliott scored the only goal of the game with a stunning long-range strike as Liverpool lifted the gloom by beating Wolves to reach the FA Cup fourth round.
Elliott was one of eight players to come into the Reds side following a chastening 3-0 Premier League defeat at Brighton and Hove Albion, and the teenager made his mark with a sublime first-half finish.
Wolves felt aggrieved not to have won the third-round tie at Anfield after having what appeared to be a late winner ruled out for offside, but they did not pose a threat in the replay at Molineux on Tuesday.
Victory for Jurgen Klopp's side was their first in four games, setting up an opportunity to exact revenge on Brighton in the fourth round.
Liverpool quickly sparked into life after the floodlights briefly went off in the first minute, and Elliott put them in front with a stunning strike.
The teenage midfielder burst forward on the break and let fly with a left-footed strike from 25 yards that flashed past Jose Sa in the 13th minute.
Kostas Tsimikas fired over the crossbar and Cody Gapko was also off target, before Adama Traore showed a complete lack of composure when he drilled wide from a tight angle in a listless first-half display from Wolves.
Julen Lopetegui made a double substitution at the break, introducing Matheus Nunes and Nelson Semedo, but Wolves continued to look short of ideas going forward.
Mohamed Salah entered the fray with 25 minutes to go in a second half that was devoid of quality, with Wolves captain Ruben Neves sending a free-kick just over the bar from a promising position.
The hosts applied some pressure but were unable to fashion an equaliser as the Reds came out on top in a forgettable all-Premier League encounter.
What does it mean? Relief for Reds
Liverpool manager Klopp vowed he would not quit and called on his players to go "back to basics" after they were played off the park by Brighton on Saturday.
A much-changed team showed the sort of intensity the German demands as the Merseyside club, who are ninth in the Premier League, set up another trip to the Amex Stadium.
Wolves must turn their attention back to climbing away from the relegation zone, and there was some good news as they confirmed the signing of Pablo Sarabia from Paris Saint-Germain before the game.
Elliott shows energy
Liverpool have looked short of legs in the middle of the park, but Elliott made a difference.
The former Fulham youngster's fourth goal of the season was worthy of winning any game, and he worked tirelessly off the ball while showing his quality in possession to surely earn a starting berth against Chelsea this weekend.
A case for the defence
Liverpool had failed to keep a clean sheet in seven games before finally keeping the back door shut at Molineux.
Wolves were unable to trouble them, with Ibrahima Konate and Joe Gomez solid at the heart of the back four.
Key Opta Facts
– Liverpool have progressed from 11 of their last 12 FA Cup third-round ties, with this their first win against Wolves in the competition since February 1952.
– The Reds have won their last four away games against Wolves in all competitions, their longest ever winning streak against them at Molineux.
– Goalkeeper Caoimhin Kelleher has been on the winning side in all five of his FA Cup matches for Liverpool, keeping his second clean sheet in the competition in this match.
– Elliott's winner was Liverpool's first FA Cup goal from outside the box since Curtis Jones' strike against Everton in January 2020. Each of their previous 24 strikes had been inside the box.
Liverpool really need a win when they face Chelsea in the Premier League on Saturday, while Wolves travel to second-placed Manchester City on Sunday.