The City Ground club had to hold on against the league’s bottom team, even as they maintained their top four quest.
Elliot Anderson, Callum Hudson-Odoi and Chris Wood had given them a clear lead, before the Saints roared back.
“We are delighted we won the game. We knew how tough it would be,” manager Nuno Espírito Santo said.
“We have to look back and see how demanding the one before this one was (Tuesday night’s 1-1 draw against Liverpool) - the high intensity required of our players, in terms of focus. We knew this could happen.
“It was a warning. It was a warning for everybody to realise that this league is very tough.
“We had been talking about how competitive this league is. You have to give your all and cannot ever underestimate or think the game is over. I credit Southampton a lot, with the way they created so many problems.
“We started really well. We pressed, we recovered and we scored, and everything becomes easier. First half we didn’t concede any situations that could create problems for Matz (Sels).
“The beginning of the second half was good and we controlled the possession. But it was a strange second half. We had a lot of stoppage, the game didn’t flow and we had VAR decisions and 12 minutes of added time. I think it always helps the team that is trying to get back in the game.
“We didn’t manage the situation so well. I’m delighted we won the game.”