The England youth star, who is now 22, marked his full debut with two goals in a 5-3 win over Cardiff City in the Carabao Cup second round on Wednesday.
After the game, he lavished praise on Martin for convincing him to join from Aston Villa.
Archer told the Daily Echo: "The plan originally was to leave again, no matter how well I did in pre-season.
"Obviously, Saints came in. I spoke with the gaffer and we had a good connection. He told me everything that I wanted to hear, to be honest."
On Martin convincing him to join, Archer responded: "I think the brains. I don't think a lot of people see that. Maybe they do.
"When you speak to him, he knows what he's talking about. Behind the scenes, a lot of coaching, and development of players. That's exactly what I wanted."