The Saints' boss was delighted with the way in which his side responded after they went a goal down, while Paul Lambert lamented his team's second-half collapse at St Mary's
Saints started the better of the sides but went behind when Darren Bent poked home from close range before the break. However, two goals from Rickie Lambert, with Nathaniel Clyne and Jason Puncheon adding one each, secured a 4-1 victory for the south coast outfit in the second-half.
The three points were the newly-promoted side's first of the campaign, after they lost their opening four games in a tough start to the season. The manager was delighted to see his side's hard work finally pay off, adding that they have embraced the challenge so far.
"The players have done ever so well, it's a great victory for us, especially as we've gone 1-0 down," the 47-year-old told reporters. "We responded in the right manner, there's a lot of belief at Southampton.
"We've just had two back-to-back promotions, we're in the Premier League now, we know it's going to be challenging we know there is going to be a lot of pressure all over the place. We embrace that challenge because this is where we want to be.
"The players went out and performed very well, Rickie Lambert's now got four goals in the league. I think the way he linked up with gaston Ramirez, who looks a talented player, [was good]."
Reports had suggested that Adkins job was under pressure heading into the match, after the club's pointless start to the season, despite facing both Manchester clubs as well as Arsenal.
However, the fans looked to show their appreciation with numerous chants and ovations for the boss when he appeared on the touchline - a gesture the manager was grateful of.
He continued: "From a personal point of view it's very nice because it demonstrates we're together as one at Southampton. Everyone's working very hard from the chairman down."
First-choice keeper Kelvin Davis was dropped for the fixture at St Mary's, after he conceded 14 goals in the opening four games, with 20-year-old Paulo Gazzaniga replacing him. The boss said it was something he had to do, despite the past performances of the veteran.
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He added: "Kelvin Davis has been outstanding since I've been at the club. He's an integral part of everything that's gone on here and a great professional, but I have to make decisions. That's what managers do."
New signing Maya Yoshida had a solid game at centre-back for Saints, which will please supporters after the 6-1 thrashing by Arsenal at the Emirates. Adkins praised the Japanese defender for his display, as well as revealing he is comfortable with speaking the language as well as performing on the pitch.
"I thought he had an outstanding game again today," he added. "The two forward players of Aston Villa are going to be a handful for anybody and he's come in with an assurance.
"He passes the ball ever so well. We've got good competition for places now so I'm delighted with his performance. His English is good, he understands the game."
Villa boss Paul Lambert was disappointed that his side lost a 1-0 lead, as they looked the better side after scoring, while indicating there was no reason for his players' heads to drop after conceding twice in quick succession.
"They shouldn't [have a problem with conceding], they've gone down before in games. You're still well in the game at 2-1 down," he stated.
"The first-half I thought we were really fine, the second-half we had too much running to do and you can't defend that deep and expect to get away with it."
The Scot also confirmed that Stephen Ireland, who was withdrawn at half-time, is suffering from pain in his arm, while suggesting his absence was a big factor in the second-half.
He added: "We need to have a look at him. His arm is giving him a problem.
"He's a good player, the first-half I thought we were a threat, and he was at the forefront of it."