Saints had to settle for a 1-1 draw when Nathan Dyer cashed in by seizing on goalkeeper Paulo Gazzaniga pass out from the back after Maya Yoshida miscontrolled.
Morgan Schneiderlin had earlier given the home side the lead at St Mary's and Adkins preferred to focus on the reaction of his players under adversity as they moved off the bottom of the Premier League.
"The reaction of everybody at the football club has been positive," said Adkins.
"Everybody is working very hard and that gave us a good defensive platform to go and pass the ball and create opportunities.
"We've got a very young side. Unfortunately, the decision process for the young goalkeeper has led to the goal but he'll learn from that and we'll take the positives and move on."
He added: "We've gone and performed well today, the supporters have again been magnificent and everyone is pulling in the same direction and because of that it gives us a great opportunity.
"We all want to be successful at Southampton and it's magnificent that the supporters have always been great with me. The players have been fantastic and we're all working together.
"We're in the toughest and most desirable league in the world and we intend to stay here and kick on from that."