Petty was the fifth quarterback selected in the draft and the first Baylor quarterback selected since the Washington Redskins took Robert Griffin III with the second overall pick in the 2012 draft.
The 6-3 Petty had been projected to be picked in the second or third round on Friday, but Colorado State quarterback Garrett Grayson and Oregon State quarterback Sean Mannion were selected ahead of him in the third round.
The Jets gave up the 104th pick in the fourth round and a seventh-round pick to the Jacksonville Jaguars to move up to take Petty.
“That was the longest 48 hours of my life,” said Petty in a teleconference with the New York media.