Manager Jurgen Klopp, watching on from the back of the first tier of the Main Stand due to his one-match touchline ban but in constant communication with coach Vitor Matos, would have enjoyed his different perspective as the forward made it seven goals in his last 10 appearances.
A fourth successive win in all competitions saw his side enter the mid-season break on an upward curve as they moved to within five points of fourth-placed Newcastle.
Roberto Firmino's sixth-minute header gave them the ideal start and although Che Adams equalised as soon as Nunez made his mark then Nathan Jones' first match in charge after replacing the sacked Ralph Hasenhuttl turned into an uphill struggle.
Five of Liverpool's starting line-up are heading to Qatar next week and two others, Ibrahima Konate (knock) and Jordan Henderson (personal reasons), were absent but it was Nunez who took the opportunity to sharpen himself up for Uruguay's challenge.
Both goals were instinctive and predatory; a first-time side-foot volley and sliding finish inside the six-yard area.
The 23-year-old has taken time to bed in to his new surroundings following his potential club-record transfer from Benfica but has noticeably gone up a gear in the last month.
It takes a certain kind of player to be able to overshadow Mohamed Salah, who had 10 in 10 before this game, but Nunez has all the attributes, albeit it different from the Egyptian, to ensure it is no longer a one-man show.
He has also extinguished all the questions about allowing Sadio Mane to depart to Bayern Munich, with his best recent performances coming in the left-sided role - again played in a completely different way to the Senegal international.
That he was helped by some rather naïve Southampton defending would not give full credit to his performance, which was direct, aggressive and clinical.
Visiting boss Jones opted for a 4-4-2 formation for his Premier League debut but his side were far too open and they were frequently carved apart.
Even Southampton's first goal at Anfield since 2015 - after Adams took advantage of Virgil van Dijk switching off to head home a ninth-minute equaliser from James Ward-Prowse's free-kick - could not paper over the cracks.
Liverpool had taken the lead three minutes earlier when, after one-time transfer target Duje Caleta-Car had pulled down Salah on the counter-attack, Firmino deposited a flicked header inside the far post from Andy Robertson's inswinging free-kick.
Nunez warmed himself up with a cross to Salah which saw Gavin Bazunu superbly repel a first-time shot, although the goalkeeper was powerless to prevent Liverpool's second.
Harvey Elliott, the 19-year-old who continues to grow in a right-sided midfield role, played in Nunez on the left of the penalty area and cleverly lifted over a pass which was volleyed home with aplomb.
The best move of the match, started by Thiago nicking the ball on the edge of Liverpool's penalty area before it was transferred through Nunez and Elliott, deserved a goal as the Uruguayan backheeled to Firmino only for his close-range shot to be charged down.
But another exquisite move did breach Southampton in the 42nd minute as Robertson raced onto Thiago's precise through-ball to cross low for Nunez to divert past Bazunu.
With that assist the Scotland captain equalled former Everton left-back Leighton Baines' record of 53 for the most Premier League assists by a defender.
So open were Southampton it appeared Liverpool could have made the score anything they wanted in the second half but to their credit the visitors rallied and Alisson Becker was the busier of the two goalkeepers.
The Brazil international stuck out a hand to deny Mohamed Elyounoussi from close range, then stopped substitute Samuel Edozie with his leg - either side of Salah having a penalty claim turned down as he bore down on Bazunu - before producing a reflex stop from another Adams header.
Veteran midfielder James Milner's introduction for his 600th Premier League appearance, only the fourth player to reach the mark in the competition's history, completed day of positives.