Cameron Archer scored for Saints, as the Gunners eventually won 3-1 on the day.
Martin told the Daily Echo: "I feel really proud of the team in a lot of ways. The willingness to defend, fight and run that wasn't there in the first half on Monday was there in bundles today.
"I feel like they've grown since Monday, they've learnt a lot. They stuck with the game plan, I thought it worked really well.
"We knew we were going to be under a bit of pressure in the first half especially. We asked them to be a bit braver in the second half, to take the ball and we did that.
"We were not in front for long enough and that's the only frustration. We have a chance to take the temperature out of the game at that point when Flynn receives the ball.
"We go back into the bodies and it's really unfortunate because Flynn's been amazing again today.
"The game goes a bit crazy and we have to stick with doing the right things at that moment. Then we're 2-1 down, trying to chase the game, not accepting just being beaten.
"We get caught with a third goal which, again, is just not good enough. The same frustration is conceding goals that are not good enough to concede against a really brilliant team.
Martin continued: "I think it's unfair on some individuals that we have conceded so many goals so far.
"We played really brilliantly at Newcastle away and against Manchester United for half an hour at home.
"They are not the games that are going to define us. The games that define us are coming up. We have three of them in the next block of four games. We have to do better in those games.
"We've had a few already against teams that are going to be in and around us and having the same fight to stay in the league or stay established. We have to take the same mentality today."