World number 38 Barthel recovered from a slow start as she saved three break points from 0-40 down in the opening game before breaking Suarez Navarro to love and consolidating the advantage to move 3-0 in front.
She held serve once more to move 4-1 ahead, but 10th seed Suarez Navarro took control from there in the first set by winning five games in a row to clinch the opener 6-4.
Both women struggled on serve in the second set, with Barthel securing the only hold on her way to a 6-2 success that forced a decider.
Suarez Navarro broke in a mammoth opening game at the start of the third and that seemed to break Barthel's resistance. The Spaniard never gave up her advantage, winning 6-3 to seal victory in two hours and 11 minutes.