The Gunners fell behind to an early Stuart Armstrong strike, but rallied to lead at the interval courtesy of Nicolas Pepe and Saka, with the England international then teeing up Alexandre Lacazette to complete the 3-1 victory.
Saka's goal was his fifth in the Premier League this season – more than all other teenagers in the division combined.
He has been impressive in recent weeks and has taken on a pivotal role in the Arsenal attack, having previously moved across the pitch to cover several different positions.
"It's a dream so I just have the passion when I get on the pitch. The boss makes everything clear for me and at the same time gives me the freedom to go and do what I want to do in the final third."
Southampton, having beaten Arsenal just three days earlier in the FA Cup fourth round, have now won just one of their last seven league outings.
Yan Valery came into the side to replace the injured Kyle Walker-Peters on Tuesday night and says lessons must be taken from the latest setback ahead of Saturday's visit of Aston Villa.
"We started the game right and scored the first goal, but after that we made a few mistakes and I think all the goals came from our mistakes," he said.
"Now we need to just learn from it and move on. In the second half we started good as well, but, when they scored the third one, we just gave up.
"We need to be ready for Saturday and go again. We have done everything right in the first half of the season, so I don't know why it wouldn't happen again in the second half."