Jason Puncheon had a first-half header ruled out, but he would not be denied as his goal after the break propelled Saints up to 15th in the Premier League table.
Adkins believes his players are now relishing the challenge of competing in the top-flight after a difficult start to the season.
"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."
Adkins was particularly pleased with Puncheon's performance, but felt the right-winger could have finished with more goals.
"It was a great goal, I am delighted with it," said Adkins.
"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."
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