Seventh-ranked Monfils, a French Open quarter-finalist, will face Russia's 45th-ranked Dmitry Tursunov later on Thursday due to a compressed schedule forced after rain postponed six matches on Wednesday at the US Open tune-up event.
Monfils will have to win five matches in four days here to claim his fourth career ATP crown after titles last year at Montpelier, in 2009 at Metz and in 2005 at Sopot. His only career outdoor hardcourt final was last year at Tokyo.
Sweeting, who won his first ATP crown at Houston in April, surrendered a break to Monfils on the Frenchman's third chance of the eighth game, the only break points either man had in the first set.
Monfils then held, capturing the set with his seventh ace after 32 minutes.
Sweeting, ranked 66th, surrendered a break to Monfils in the third game of the second set but broke back to 3-3. Both held to a tie-breaker, in which Monfils raced to a 6-1 lead and finished off the American after 85 minutes.
Tursunov needed only 77 minutes to oust Flavio Cipolla 6-1, 6-4.
Monfils, 24, split with Australian coach Roger Rasheed after Wimbledon and came only with fitness coach Patrick Chamagne to Washington, where he reached the semi-finals in 2007 in his only prior appearance.