Edmund improved to 1-1 in career finals.
Gulbis had won all six of his previous ATP finals but this was his first since 2014, having beaten top-seeded John Isner in the semi-finals.
Not since 2005, when Greg Rusedski won in Newport, Rhode Island, has a Briton other than Andy Murray won an ATP singles title. "He had a great week", Gulbis said.
Meanwhile, at the Kremlin Cup in Moscow, Russian Karen Khachanov, world ranked 26, secured his third title with a 6-2 6-2 win over Frenchman Adrian Mannarino, the world number 49. "I'm just so happy". You always remember this one.
This week at the European Open though, Edmund set himself up for yet another shot at a win. Gaël made me work for it today, that's for sure, so credit to him.
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. Monfils dropped to 7-21 in career finals. "I'm just so happy".
There was a home victor at the Kremlin Cup as Russian Karen Khachanov defeated France's Adrian Mannarino 6-2 6-2 in less than an hour to claim his third ATP Tour crown.