Abdoulaye Doucoure gave Everton the lead in normal time before that goal was canceled out by Taylor Harwood-Bellis.
Saints goalkeeper Alex McCarthy was the hero in the shootout as he denied Ashley Young to clinch the win.
Martin, asked if this could kickstart their season, said: "I hope so. We'll see on 5pm on Saturday.
"I feel like all you can ever ask is for the players to analyse the performance from Saturday, try and learn from it - to take something from it.
"What we learned was actually for half an hour we were brilliant (against Manchester United). The best performance of the season so far and looked very much like the team we wanted.
"Then we have a disappointing moment and it completely derails us. Tonight we didn't. We had a tough moment conceding and then they responded brilliantly.
"I think that's the learning and you analyse that and you improve and you grow from that and they did that tonight.
"We've hurt ourselves a lot in the first four games and a lot of that is down to a bit of anxiety and a bit of fear of the Premier League for some of the guys going into it for the first time.
"Some of the guys were hurt by it last time but I feel like we're going to grow through that and we just need that first result.
"Once we get that, I'm really excited about what the group is capable of achieving because we have a young, hungry group that is willing to do brilliant things."