The defensive tackle is expected to fill in for Jay Ratliff, who will miss the first six weeks of the season after being placed on the 'physically unable to perform' list.
The 27-year-old told reporters: "It's a great opportunity for me. Being out last season like you guys already said, I've just got to run with the opportunity and just show this team what I've got and why they picked me up.
"You never know when this game is going to end, that's every guy in the locker room. You just don't know. That's why you've got to take full advantage of every opportunity, and I'm just glad they gave me the opportunity.
"Little things have helped me out so much this year just technique-wise, getting off the ball, pass rush. It's hard to learn this stuff, but [defensive line coach Rod Marinelli is] such a great coach teaching the little things."
Hayden joined the Cowboys last season, but missed the entire campaign with an ankle injury.