The West Virginia quarterback had been rated as the top QB among this year's college prospects, but he had to wait until the 39th pick before being drafted.
Speaking at a press conference, the 22-year-old said: "It's been great. A great experience, a dream come true, to finally hear my name being called and to be a Jet is one of the things that I will carry with me for a lifetime.
"From sitting in the green room for so long, thinking that you may go in the first round, it was kind of tough to stomach, but I was able to go out and have dinner with my mom and just refresh myself, clear my mind, and wake up the next day with a whole new mindset.
"There's always going to be an adjustment period. Right now, my focus is just getting into that playbook and learning the terminology, making sure that I have that down first."
Smith is expected to compete with Mark Sanchez to be the Jets starter for the 2013 season.