The match was suspended because of rain Tuesday with Monfils leading 4-2 in the first set. After clinching the set when the match resumed Wednesday, the Frenchman had two match points in the second-set tiebreaker, but Hanescu saved them and went on to complete his first win over a top-10 player after 11 losses.
Monfils reached the final of the tournament last year, only to be forced to drop out of the match against Albert Montanes of Spain because of injury.
Montanes also lost Wednesday, going down 2-6, 6-4, 6-3 to fellow Spaniard Pablo Andujar.
Six of the eight seeded players have lost in the first round.