After breaking Brazilian fans' hearts by topping Mariana Silva in the semi-finals, top seed Trstenjak handily beat Agbegnenou by ippon less than two minutes into the final.
Israel's Yarden Gerbi and Anicka van Emden of the Netherlands won bronze.
Agbegnenou's silver snapped a medal drought for France, seen as a top contender in Rio but which had come up short in the first three days of judo action.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
France is one of just three countries fielding a full team of 14 judokas in Rio and was hoping for a repeat of its strong performance in London four years ago, when it claimed seven medals - tied for the most with Japan.
(Reporting by Chris Gallagher; Editing by Bill Rigby)
