Jurgen Klopp made nine changes to the XI that lost by the same scoreline to title rivals Manchester City last Thursday and they turned in a lethargic first-half display, trailing at the break to Raul Jimenez's opener.
Divock Origi found a fine equaliser but Portugal midfielder Neves raised the decibel levels significantly at Molineux with a decisive 55th-minute long ranger.
One of the players Klopp retained from the City defeat, Croatia centre-back Dejan Lovren, pulled up and had to be replaced in the sixth minute, meaning 16-year-old Dutch defender Ki-Jana Hoever became Liverpool's youngest player in FA Cup history
It was a couple of senior men who let their manager down in the 38th minute when Diogo Jota stole possession from James Milner and burst past Fabinho.
Jota played in Jimenez, who still had plenty to do but bore down on goal and steered an unerring finish beyond Simon Mignolet.
Liverpool pulled level six minutes into the second half when the hitherto anonymous Origi engineered a fraction of space on the edge of the box to smash home left footed.
Parity did not last long as Neves found an even better finish by launching a typically dipping and swerving effort from 30 yards.
Jimenez had the chance to move Wolves closer to round four when he picked Hoever's pocket but could not adjust his feet after going around Mignolet.
The run of spectacular second-half strikes continued as a sumptuous free-kick from Liverpool winger Xherdan Shaqiri hit the inside of the right post and somehow bounced to safety.
Klopp then called for the big guns, sending on Mohamed Salah and Roberto Firmino, but Wolves held on.
What does it mean? Klopp turns attention back towards biggest challenge
Manchester City manager Pep Guardiola insisted he was not in a position to choose between competitions and named a strong side to demolish Rotherham United 7-0 on Sunday. Klopp, whose squad depth is improved but does not quite match that enjoyed by Guardiola, evidently does not feel the same way and such a result always felt likely after the team-sheets were handed in.
Neves adds another stunner to his collection
Former Porto captain Neves won rave reviews early in the season but has only started one of Wolves' past three Premier League matches. The 21-year-old was back to his sublime best here, controlling midfield with his exhaustive range of passing before producing another goal to linger in the memory.
Listless Sturridge fails to make his case
Liverpool's front two were notable for operating in not-so-splendid isolation before half-time but Origi at least got himself on the scoresheet in style after the break. Daniel Sturridge, who was England's first-choice striker at the start of Gareth Southgate's reign, appears to have accepted his lot as back-up following a string of injuries and the fabulous form of Jurgen Klopp's forwards. However, he only has six months remaining on his Anfield contract and this was a sorry pitch for an extension.
Key Opta Facts:
- Wolves earned their first home win against Liverpool in seven attempts (D3 L3), since a 1-0 win in August 1981.
- Liverpool have been eliminated from six of their last eight FA Cup ties against fellow Premier League opponents.
- Wolves have eliminated Liverpool from the FA Cup five times – Stoke City are the only side they have knocked out of the competition more (6).
- Raul Jimenez's goal was Wolves' first at home against Liverpool in 398 minutes, since Kenny Miller scored a 90th-minute goal against them in January 2004.
- Eight of Ruben Neves' nine goals for Wolves in all competitions have been from outside the box, with his other strike coming from the penalty spot.
- Liverpool's Divock Origi scored his second goal in the FA Cup, with the other also coming against Wolves back in January 2017.
- Liverpool's Xherdan Shaqiri has ended on the losing side in all four of his appearances in the FA Cup (three times with Stoke).
Liverpool will hope for more of the same from Wolves when they travel to Manchester City in seven days' time, by which point an away victory at Brighton and Hove Albion could again put the Reds seven points clear of the reigning champions.