The in-form Uruguay forward had been on the pitch for just 10 minutes when he collected a pass from fellow replacement Trent Alexander-Arnold and unleashed a thunderous finish into the top right corner.
Cody Gakpo gave Jurgen Klopp's much-changed visitors a first-half lead before Justin Kluivert's maiden Cherries goal levelled proceedings amid treacherous conditions caused by Storm Ciaran.
Parity lasted just six minutes as Nunez's third goal in as many games emphatically earned the nine-time winners a place in the last eight to ensure an entertaining fourth-round tie would not be settled by the jeopardy of a penalty shoot-out.
Mohamed Salah was one of only three Liverpool players retained from the team which began Sunday's 3-0 win over Nottingham Forest and captained an experimental line-up.
Bournemouth, buoyed by beating Burnley 2-1 to register their first Premier League win under head coach Andoni Iraola, made just three changes and had the better of the early chances.
The recalled Kluivert was twice denied by Reds goalkeeper Caoimhin Kelleher, while the unmarked Ryan Christie wastefully side-footed wide following Antoine Semenyo's cut-back.
Liverpool were dominating possession and should have taken the lead in the 27th minute when Harvey Elliott tamely fired straight at Cherries keeper Andrei Radu from level with the penalty spot.
The lively midfielder made amends four minutes later with a key role in the opener.
After Kostas Tsimikas' corner was partially cleared, Elliott's fizzing effort was back-heeled goalwards by Gakpo and then finished at the second attempt after Radu repelled the initial effort.
Bournemouth boss Iraola said pre-match his side faced a "big challenge" but also had a "big chance" to reach the next round.
In relentless driving rain, his team began the second period on the front foot and almost received a helping hand from the swirling wind as Alex Scott's inswinging corner from the left was headed off the line by Joe Gomez at the back post.
Salah nodded narrowly wide as Liverpool threatened to extend their lead before Semenyo sliced past the right post at the other end and substitute Marcus Tavernier tested Kelleher from distance.
Bournemouth deservedly levelled in the 64th minute.
Another Scott corner caused problems for the visitors' static defence, leaving the unmarked Kluivert with a simple headed finish to claim his first goal in English football.
The Cherries looked well-placed to push for a winner but were soon behind again.
Nunez initially lost control of Alexander-Arnold's sweeping pass but atoned in devastating fashion by cutting inside Chris Mepham from the left wing to thump high into the net from just inside the hosts' 18-yard box.
Bournemouth were given five minutes of added time to force spot-kicks but rarely threatened a second equaliser as Klopp's men comfortably held on to progress.
Additional reporting from PA