The Stadlers birdied the 18th in regulation to match the Irwins with 12-under 60s on Sunday. The families, who shared the overnight lead, finished 36 holes tied at 24-under-par 120.
The Stadlers were first off the tee in the extra session and both found the fairway. Steve Irwin hooked his tee shot into a pond off the 18th at the Ocean Club Golf Course at Atlantis Resort but father Hale landed in the fairway.
In the scramble format, the teams elect which shot they want to play but the Irwins were obviously left with only one option. Steve Irwin played first and knocked his approach to 15 feet while Hale hit an indifferent shot that found the middle of the green. Both Stadlers pulled their second shots but Kevin found the left edge of the green, 30 feet from the cup.
Kevin Stadler left his birdie try one rotation short of dropping but Craig did not. His putt tracked toward the center of the cup and fell in to put the pressure on the Irwins.
Steve Irwin putted first and skirted the hole but Hale's putt went left off the face and never grazed the hole.
'I still don't know what the heck happened,' said Craig Stadler. 'That's the only putt I've made in two days that was over a foot and a half. I wasn't expecting a whole lot.
'I am so proud of Kevin. He played so well,' said Craig Stadler. 'I did absolutely nothing these two days. I didn't have to. He played well. I just rode his butt all day.'
The Stadlers collected $200,000 for their winning effort.
Johnny and John Miller, Jr., shared third place with Raymond Floyd and his son Raymond, Jr. The pair each fired 12-under-par 60s on Sunday and finished the two-day scramble event at 22-under par.
Gary and Wayne Player, Hubert and Myatt Green and Dave and Ron Stockton shared fifth place at minus-19. Jack Nicklaus and Jack, Jr., tied for eighth with Larry and Drew Nelson and Tom and David Kite at 17-under-par 127.
Final scores from the Father-Son Challenge