Defending champions Raymond and Robert Floyd share third place with Larry and Josh Nelson at 9-under-par, while the Miller family, Johnny and Scott, are alone in fifth at minus 8.
The Irwins used a stretch in the middle of the round where they went 10-under-par over 11 holes to jump to the top of the leaderboard. That run left the Irwins in first by themselves with no other father-son duo threatening.
'Once we got on that roll, we just kept going,' said the elder Irwin.
This marks the third time the Irwins have opened the Father-Son Challenge with a 59. In 1998, the tandem shot a 65 in the second round and finished third while in 1999, they posted a 68 on Sunday and tied for sixth.
'In the past, I think I would try to get too cute with my shots,' said Irwin. 'Steve always hits first and when he hits a good shot, I try to be too precise instead of hitting my normal shot. I think I will just try to play my normal game and keep things simple.'
The Stocktons, Greens and Charles family are tied for sixth with first-round 65s.
Tom and Michael Watson and Tom and David Kite are knotted at 66.
This event features professionals that won major championships in their career. It is a two-day event that is contested in a scramble format.
Jack and Jackie Nicklaus as well as Jerry and Wesley Pate withdrew before the tournament started due to injuries to the fathers.