It was the first match of the year for Sun who has been trying to recover from a wrist injury. Fruhvirtova had come through qualifying winning three matches.
Twenty-four-year-old Sun held significant leads in both sets, 3-1 in the first, after breaking her opponents serve in the first game of the match and again 3-1 in the second.
The left-handed New Zealander, ranked 86th, looked to fight back in the second set after being down 3-5 and held off several match points before winning her own serve and breaking her opponents to level the scores 5-5.
However, Fruhvirtova ranked 132 then broke Sun's serve to go ahead 6-5 and then held her own serve to win the match.
Erin Routliffe and her new doubles partner Asia Muhammad are in action on Wednesday afternoon.
