An up and down first four months of the season ended on an upward curve as Nunez scored twice after Che Adams has cancelled out Roberto Firmino's sixth-minute goal to secure a fourth successive win in all competitions.
The top-four is now within touching distance, which is an improvement on a couple of weeks ago when back-to-back defeats against Nottingham Forest and Leeds raised serious questions about their ability to compete for a place among Europe's elite.
A five-week break will now provide time for consolidation before thoughts turn to the second half of the campaign and, like last January when they made a snap decision to bring in Luis Diaz, Klopp has not ruled out making signings.
"We are always open to these kind of things, but we don't have to open this discussion now," said the German, who has been criticised for not strengthening his midfield options.
"If there is something possible we will do it but the door is open and we are open as well.
"It is not about wanting. We want to improve the team all the time, we just believe in the training ground we can do it there."
Klopp had to watch the game from the back row of the Main Stand's bottom tier as he served a one-match ban for abusing assistant referee Gary Beswick in last month's win over Manchester City.
From there he saw Nunez continue his improvement as he took his tally to seven goals in 10 appearances as his team won the game in the opening 45 minutes.
"First half was really good and the second half was not, we made a lot of mistakes," he said.
"First half we did everything Southampton didn't want us to, used the speed of Mo (Mohamed Salah) and Darwin, the spaces between the lines.
"In the system Darwin has to defend the left wing but offensive he has to be more in the centre.
"Second half we became too passive in defending, last line didn't push up and it was too much space until they faced our last line and they had then their moments."
For new Saints manager Nathan Jones it was a tough baptism to life as a Premier League manager.
But he insists he did not leave Championship side Luton just to enjoy the top-flight experience.
"I don't just want to be a Premier League manager, I want to make sure I am successful at the level," he said
"It was a very proud moment and it would have been easy for me not to take the game today.
"I only had one day to work with them, but that is not my character, my style.
"I wanted to work with them, to get to know my players, my staff. It is one of the toughest places you can start at but I have learnt a lot.
"The goal just before half-time killed the game off but second half I was really proud of how we reacted. Alisson (Becker) made three wonderful saves to deny us."