The 22-year-old has risen to number one in the world following a string of victories and consistent performances over the past year.
McIlroy also achieved the goal of winning his first Major title at the US Open at Congressional last June.
Dawson has said that it is great for British and European golf to have the next big superstar.
"You're really seeing the old guard in Tiger, he's only mid-30s, isn't he, and the young Rory," said Dawson. "Every generation has its stars and Rory is going to be this one.
"It's very gratifying to see someone we've seen from his amateur days and played in a lot of events come through like this.
"It's a wonderful time for British golf and European golf, having so many players from these islands and the continent of Europe doing so well."
McIlroy has one win to his name from four appearances so far this season.