Stephens broke Australian-born Robson in the first game of the match then held serve to take the first set in 45 minutes.
The 19-year-old eighth seed broke the Englishwoman in the third game of the second before Robson broke back in the sixth for 3-3.
In a tiebreaker, she shot out to a 5-1 lead that Robson cut to 5-4 before Stephens converted her first match point on serve.
World No.38 Stephens was ousted in the quarter-finals of the Brisbane International by her idol and eventual champion Serena Williams last week.
There were high hopes for 18-year-old Robson in her first Hobart tournament after she became the first Brit to make a WTA final in 22 years at Guangzhou last year.
Stephens said a breakthrough tournament victory was not her only focus in Hobart.
“The beginning of the year’s always crucial for any player to get your bearings and just making a ranking jump last week I think that’s pretty crucial,” she said.
“No tournament is easy.
“I’m playing pretty well and I have to keep doing what I’m doing and then hopefully a win will come.
“I’m not getting any younger so I need to hurry up and win something.
“Who knows? It could come this week it could come at the last tournament of the year.”
Stephens revealed another ambition when a junior player hit her with three questions.
Asked what she would do if she wasn’t a professional tennis player, she said: “My family owns a funeral home.
“I want to be an embalmer but you have to go to a lot of school for that.
“So maybe just work in the funeral home.”
In the other first round match played on Sunday, fifth seed Carla Suarez Navarro was ousted by Belarusian world No.57 Olga Govortsova 6-2 6-1.
Brisbane international finalist Anastasia Pavlyuchenkova pulled out of the tournament with a hip injury.
By David Beniuk HOBART, Jan 6 AAP –