The eighth seed Stephens had enjoyed a run to the quarter-finals in Brisbane the previous week, only beaten by eventual champion Serena Williams.
And she proved too strong for Robson, ranked 15 places below her at 53 in the world, holding off the Brit in two tight sets.
In the other game of the day in Tasmania Olga Govortsova stunned fifth seed Carla Suarez Navarro with a comprehensive 6-2 6-1 win.
WTA Hobart results
8-Sloane Stephens (US) beat Laura Robson (Britain) 6-4 7-6(4)
Olga Govortsova (Belarus) beat 5-Carla Suarez Navarro (Spain) 6-2 6-1