Newly-promoted Southampton were beaten 6-1 by the Gunners in the reverse fixture at the Emirates Stadium in September.
Arsene Wenger's team needed a Guly do Prado own goal to share the points at St Mary's after Gaston Ramirez had put the hosts ahead.
The result was enough to lift Saints out of the relegation zone in the Barclays Premier League and Adkins said: "It just shows things are close at the bottom. One victory or result and it's going to keep fluctuating.
"The performance on the back of the Stoke performance demonstrates everyone's working hard and fighting for each other.
"It's only two defeats in 10 games, which is important, but look at how far we've come.
"We got a good hiding down at the Emirates at the start of the campaign - six against us.
"We've got a growing maturity about the players and the performance underlines how far we have come in growing into the Premier League.
"When you look at it Arsenal have been beating teams out of sight of late but the goal was a deflection and Artur Boruc's not had anything to do in goal from that point of view.
"We had a good shape about ourselves and we were tactically bang on in restricting Arsenal to limited opportunities and when we had the ball we created good opportunities ourselves."