The victory took Liverpool into sixth place in the league standings on 22 points after 14 matches although Brighton & Hove Albion can climb back above them when they play Aston Villa on Sunday.
Brazilian Firmino put Liverpool in front in the sixth minute by glancing an Andy Robertson free kick into the net.
Moments later Southampton replied with a similar effort, Che Adams rising to head in James Ward-Prowse’s set-piece delivery and score the club’s first goal under new manager Nathan Jones.
Uruguay international Nunez restored Liverpool’s lead in the 21st minute by volleying home a cross from 19-year-old midfielder Harvey Elliot.
The striker, who cost Liverpool a total of 95 million euros ($98.34 million) when he signed from Benfica in June, grabbed his second goal three minutes before halftime, meeting another cross from Robertson from close range.
The second half lacked the frenetic nature of the first although Liverpool goalkeeper Alisson Becker had to make two good saves to preserve his side’s two-goal cushion while Mohamed Salah spurned a great chance when he robbed possession and raced free into the area only to miscontrol the ball.