Edmund, the top seed at the tournament in Antwerp, defeated fourth-seeded Frenchman Gasquet 6-3 6-4 in an hour and 16 minutes.
Gasquet's compatriot Monfils, the sixth seed, had earlier beaten Argentina's second-seeded Diego Schwartzmann 6-3 6-4.
Edmund is looking to secure his first ATP title.
The 23-year-old, who had been handed a walkover into the last four when his scheduled quarter-final opponent Ilya Ivashka was forced to withdraw with a thigh injury, broke Gasquet in the second game of the first set en route to going 3-0 up.
The second set then saw him break in the third as he wrapped up the win.
Monfils defeated Argentinian Diego Schwartzman by the same scoreline to make sure of his place in the final of the Belgium tournament.
Mannarino makes Kremlin Cup final
Adrian Mannarino beat world number 46 Andreas Seppi to set up a final in the Kremlin Cup in Moscow against Russian Karen Khachanov.
The Frenchman, ranked 49th in the world, won in straight sets, 7-5, 7-5 against Italian Seppi.
Khachanov came through in three sets, 6-1 6-7 (5/7) 6-3 against fellow countryman Daniil Medvedev.
Gulbis claims Isner scalp in Stockholm
Top seed John Isner was knocked out of the semi-finals of the Stockholm Open by world number 145 Ernests Gulbis.
Latvian Gulbis came back after losing the first set 1-6 to win the next two 6-3 and reach his first final since 2014.
He will face Greece's Stefanos Tsitsipas, who had earlier defeated Italy's Fabio Fognini 6-3 6-2.
