The 21-year-old led by a set and 4-2 but could not convert his lead into victory and Gasquet came through 3-6 7-5 6-2 in Antwerp.
Edmund was playing in his first ATP semi-final, having lost quarter-finals in Doha, London and Beijing this year.
He is still set to break into the top 40 for the first time on Monday.
That would see the Englishman join Australian Nick Kyrgios and Germany's Alexander Zverev as the only players aged 21 or under that high in the world rankings.
Gasquet will face Diego Schwartzman in the Antwerp final after the Argentine saved two match points on his way to beating Belgian top seed David Goffin 7-5 2-6 7-5.
In the doubles semi-final, Britain's Dom Inglot and Brazilian Andre Sa lost 6-3 6-2 to French top seeds Pierre-Hugues Herbert and Nicolas Mahut.
Britain's Kyle Edmund missed out on a first ATP final as French third seed Richard Gasquet fought back to win their semi-final at the European Open.