The heavy loss on the south coast followed the 5-1 hammering at the hands of mid-table Serie A side Atalanta in midweek as Everton's miserable start to a season that promised so much took another downward turn.
Everton's majority shareholder Farhad Moshiri said on Friday the club were "very close" to naming a permanent replacement for Ronald Koeman, who was sacked in October.
Yet Unsworth, who has repeatedly said he wants the job while overseeing team affairs on a temporary basis, said he still did not know who that appointment would be.
"I went up to see the board of directors before the game and spoke to the chairman," Unsworth said. "You are asking the wrong guy. I do not know.
"Big decisions have to be made, not just on the manager, but players as well.
"If it needs injured players to come back for things to get better, if it needs January to go and get new players, something has to change, because this group of players are underperforming, whether it is myself or Ronald in charge."
Gyfli Sigurdsson had cancelled out Dusan Tadic's opener for Southampton in spectacular fashion on the stroke of halftime but the Premier League's leakiest defence then gave up three second-half goals as the Saints cruised to victory.
"I wouldn't go that far," Unsworth said, when it was suggested Everton were now in a relegation battle as they sat 16th, two points above 18th-placed West Ham United.
"We are in a tough moment. If you ask me that question in a week, I might be able to give you a better answer.
"The second and third goals were really poor. Teams are not having to work hard to score against us.
"Whether we play two centre backs, three centre backs, a back four or a back five, individuals are underperforming. We are conceding far too many goals. Things have to change."
(Reporting by Peter Hall; Editing by Ian Chadband)
