The Black Cats capitulated horribly after going 1-0 down on the half hour, eventually losing comfortably to the EFL Cup finalists.
However, Moyes insisted that the opener, which appeared to go in off striker Manolo Gabbiadini's arm, should not have been allowed by referee Paul Tierney.
The Scot said: "Actually, we had played well for 20, 25 minutes, maybe as well as we had played here this season at times without making enough opportunities.
"Look, the goal goes in - and the first goal is really important because Southampton are a team who have been struggling for results, they had lost five or six of their games recently, so the first goal in today's game was going to be really important for both teams.
"The boy puts it in with his arm
It doesn't touch anybody's head, it doesn't do anything like that, it just goes straight in off his arm
It's a great cross in, I must say, by Bertrand and it's difficult for the referee maybe to see it.
"But it goes in off his arm, not any other part of his body, so that shouldn't have been given and it really changes the way it goes."
If there was a debate over the first goal, there was none over the second as £14.6million January signing Gabbiadini turned John O'Shea to double the visitors' lead.
Moyes made changes in an effort to rescue the game, but Jason Denayer's late own goal and a fourth in stoppage time from substitute Shane Long completed a miserable afternoon for the home side.
Saints boss Claude Puel was delighted with his team's display, which earned just a second league win in eight attempts, and will now hand his players four days off before beginning preparations for their Wembley showdown with Manchester United on February 26.
Puel said: "It was a fantastic game and good, of course, with all those games lost before, it was very important to react and to give a good answer
I hope it's the first game of a good cycle of course.
"It was important to win this game, a fantastic performance
Now we will take four days off with the family because we deserve this with all these games
It's a good thing to do it after the win and then we can prepare for the next game, a little game.
"This game, we know, is a fantastic, exciting game
We are not favourites, of course, but we know that in football, all is possible - we saw this against Liverpool.
"To win with this result, four goals and with a clean sheet, it's very good for the confidence and good preparation also for this game."
Asked about Gabbiadini's start to his career in England, which now amounts to three goals in two games, Puel added: "It's a good start, a very good start, of course
He is a technical player
You can find him between the lines because he has quality, good control, good passing."
Southampton's mini-break will have a very different focus to that of Sunderland as they head off on a four-day team bonding trip to New York.
However, Moyes insisted the defeat would not change his plans.
He said: "It doesn't change anything
As a manager, you make decisions, you try to lead your team
You lead your football club, you try to do the right things and you have to stand by your decisions."
Source : PA
Source: PA