Pettersson has a win and two runner-up finishes on the PGA Tour this year, and last week he challenged for the PGA Championship before finishing tied for third. It was the best showing of his career at a major tournament.
He has a solid history at this event, having finished in the top five in three of the last five years.
U.S. Open winner Webb Simpson, the only one of the four major champions from 2012 in the field, shot a 66.
This is the final tournament at which players can earn points that will get them into the tour's four-event playoff run.
Only the top 125 players on that list qualify for the first of the playoff events.
Mathis is currently 136th on the FedEx Cup points list and he put himself in excellent position to become a playoff qualifier.
John Daly, who began the week in 137th place, opened with a 72 and likely needs to turn in at least a 67 on Friday to make the cut.