Goffin, who lost in last year's final, has been on a winning run in Asia. Last weekend, he claimed his first title in three years in Shenzhen, China.
Gasquet was leading 3-2 in the first when the roof was closed in anticipation of heavy rain, and Goffin struggled to regain his rhythm after the 15-minute delay.
"It's never easy to start again," Goffin said. "I needed one or two games just to get used to the conditions, then I started to feel the ball really well, feel my rhythm, become more aggressive game after game."
Gasquet had three set points on serve at 40-0 in the ninth game, but the Belgian won that and the next six games.
The win moved Goffin to No. 11 in the ATP Race to London rankings. With Novak Djokovic and Stan Wawrinka out for the season, 10th place will be enough to secure a place at the season-ending tournament.
Cilic had 11 aces and dropped only one point on his first serve.
"My serving was exceptionally good, I played really aggressive after my serves and I managed to return well a lot of good first serves from Ryan," Cilic said. "To play like this is a very big boost to my confidence."
Earlier, Schwartzman defeated Sam Johnson 6-0, 7-5 for his career-high 32nd win of the season.
"In the beginning it was really nice for me," Schwartzman said. "Then he started to serve better, got his first serve in, and became more aggressive, coming to the net a few times more. I was on the defensive many times and couldn't dominate."
In a late match, Mannarino beat Yuichi Sugita 6-2, 6-4.