Goals from Nicolas Pepe, Bukayo Saka and Alexandre Lacazette helped the Gunners come from behind and claim all three points at St Mary's, where they lost 1-0 in the FA Cup just three days earlier.
The result moved Arsenal above Southampton and into eighth place in the Premier League table, and Arteta indicated his side's positive start to 2021 could soon be boosted further by Odegaard's arrival.
"I think we are pretty close but it's not finalised as I understand it," Arteta said, when asked about the 22-year-old Norwegian playmaker.
"I haven't spoken to [Arsenal technical director Edu] or the club in the last few hours but we're very positive that we'll finalise the deal."
After a dismal run of one victory in 10 league matches between October and December, Arsenal are unbeaten in their past five Premier League games and have won four of them.
Arteta indicated that the upturn in form has restored belief to his squad, saying: "Football games are always judged by the result and that affects the confidence.
"The moment that we started to win and confidence went up and we started to grow and understand each other better, we got some players back and the spirit started to lift a little bit, things were much better to be fair."
Southampton boss Ralph Hasenhuttl lambasted his side's defending after they succumbed to a third defeat in six league games.
Hasenhuttl told BT Sport: "It's always frustrating when you concede goals, but when it is that easy I think it hurts even more.
"The first was a mistake in the build-up, but then still we had the chance to defend it and we didn't do it, and then the second one was similar.
"It's not the way we can defend, we can do it much better and we know this. After being down one at half-time it's difficult to come back. We had chances to do it, but also the final quality was not good enough.
"We know we are a good side when we play a perfect game, but today we were far off this."