The 51-year-old Dawson had seven birdies and a bogey at The Woodlands Country Club. He won the Conquistadores Classic in March in Tucson, Arizona, for his first victory on the 50-and-over tour.
''I don't know that I feel any different as far as playing,'' Dawson said. ''I don't take any more chances than I used to before I won. That was then. This is just a new week. You still have to play, make the putts.''
The 56-year-old Allen also had seven birdies and a bogey. He won twice last season.
''When you're kind of making some putts or you feel at least good with the putter, you kind of feel like you can give yourself a chance,'' Allen said. ''You don't have to be knocking down every pin to do it.''
''You just want to have a chance to win going into the last nine,'' Durant said. ''Got close a couple of times. We didn't quite close the deal. Playing last week and winning with Billy certainly helps a lot.''
Defending champion Bernhard Langer opened with a 71.
Fred Couples, second last year, withdrew from the tournament.