The 19-year-old Del Potro outlasted the second-seeded Frenchman 6-4, 7-5.
Del Potro had reached the semifinal round in previous tournaments, but had never before played a championship match on tour.
"It's like a dream for me," the young player said. "I hope to win more titles in my career, but the first one, I think is the most important."
Gasquet is 5-6 in title matches.
"I played not my best match," he said. "I was frustrated because I was playing badly, but he played well."