Koeman's men were denied a battling point a minute from time as Alexis Sanchez's close-range finish finally broke their stubborn resistance at the Emirates.
Southampton goalkeeper Fraser Forster certainly did not deserve to lose and saved well from Danny Welbeck in an even first half.
His second-half performance really caught the eye though, as he single-handedly kept his side in the game when they were pushed further and further back. He saved superbly to deny Welbeck and Olivier Giroud before being finally beaten.
Despite losing back-to-back league games for the first time this season, Koeman felt the injury luck his side had during the game was the difference between the sides.
Both Jack Cork and Dusan Tadic had to be substituted before the end, while Toby Alderweireld left the pitch minutes before Arsenal's breakthrough, but Southampton had used all their substitutions and Koeman was in no doubt how important that was.
Koeman said via Southampton's official Twitter account: "I think we had a good reaction after last Sunday.
"I think our ball possession was much better than last Sunday. We controlled the game. I think we had some good opportunities, the best of the first half off Graziano.
"Ok after one hour it was more difficult. They had some changes in their attack and it was more difficult but Fraser was outstanding today in the second half.
"The difference was after that we were unlucky after our third change in the team because we had to change Toby and continue with 10 and I think in my opinion that was the difference between Arsenal and us today."
Koeman's side remain in third place and showed plenty to suggest that they can compete with the elite sides in the league.
