Saints came into the match a place, and three points, better off than their opponents, who pipped the Hampshire side to the npower Championship title last season.
Adkins' side exacted revenge for that with Jason Puncheon's solitary goal enough to secure victory and propel Southampton up to 15th in the Barclays Premier League standings.
"It is a good three points for us," the Saints boss said. "It is important to win our home games, especially against the teams close to us in the league. I thought we thoroughly deserved the victory and, in fact, I would have liked us to have scored some more goals from the opportunities we created.
"All-in-all, we've got to be happy with the clean sheet and three points. The players stepped up to the mark.
"There might be talk [of pressure] on the outside but the players deserve all the credit. They work really hard on the training ground and there is a great togetherness about them.
"They've gone out and put on a very good performance. I think if we can keep concentrating on our performances and what we want to do, we give ourselves an opportunity to go and win."
While there were many positives for Southampton, including a second Premier League clean sheet, they did lose captain Adam Lallana to injury in the first half.
"We've got him in a brace and he is on crutches at the moment," Adkins said. "That is standard procedure.
"He has got a knee injury and what we will do is assess that over the next 24 to 48 hours and that will give us a better prognosis to how long potentially he will be out for. It could be short, it could be longer. We don't know so we won't guess at the moment."