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 today, 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 today 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."
Right-winger Puncheon was the standout player for Southampton and Adkins felt he should have had two goals today, having seen a first-half header chalked off.
"It was a great goal, I am delighted with it," Adkins said of the goal that did stand.
"Jason could have had a few more. He had a few opportunities and a perfectly good goal disallowed as well.
"But the goal is off the training ground and we have a midfield player dropping out in between the defender and the full-back.
"We get somebody in the pocket, our full-back playing up high and wide player coming inside. Good quick play and an excellent goal."
Asked if he thought the disallowed goal was one of a number of decisions to have gone against Saints recently, Adkins retorted: "Yes, but it will change.
"We will keep grafting, we won't whinge about it and we will get on with it."
While there were many positives for Southampton today, 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."
Southampton fans will hope the two-week break they now have due to scheduled opponents Chelsea's Club World Cup commitments will allow Lallana time to recover.
Adkins also revealed that the period should be enough for Argentinian goalkeeper Paulo Gazzaniga to return to training from a shoulder injury.
While Southampton can relax, starting with their Christmas party tomorrow night, Reading face a quick turnaround.
Brian McDermott's side were second best at St Mary's today as they slumped to a fourth successive defeat - hardly ideal preparation for a trip to Sunderland on Tuesday.
"We are disappointed," the Reading boss said. "We had an opportunity [to win] in the first half and hit the post.
"That first goal was always going to be really, really important. They got it.
"In the second half I didn't really feel we stepped on the way we needed to get three points, never mind one.
"I think it is hard to look and reflect straight after the game. At the moment, you would imagine there would be a lot of doom and gloom around the place after a loss like that.
"I understand that completely because the second half we did not really do ourselves justice, especially with the ball.
"We know we have to be better than that and we now have another game on Tuesday to pick ourselves up."