The 21-year-old Republic of Ireland forward secured a move to the south Wales club for roughly £1.5 million.
He is now under the guidance of manager Russell Martin, who played alongside Saints winger Redmond at Norwich from 2013 to 2016.
Speaking to the club website, Obafemi revealed: "Nathan Redmond spoke to me as he played with the coach at Norwich.
"He said he would look after me, and that I should go and show the world what I can do, which is what I should do and it's about time I do that.
"I am looking forward to the future, I was looking forward to coming here in January but now I am here permanently I want to show what I can do. I am looking forward to playing for all the Jacks.
Obafemi added: "It feels good to have the number nine shirt and I want to create my own legacy for it. I can remember watching some great players up front for Swansea.
"I loved Wilfried Bony, he was a great player and I would love to leave my mark here in the way he did, by scoring goals and helping the team be successful."
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