Solomon Dacres beat Marin Mindoljevic at York Hall in London after the aggregate score was tied at 5-5.
Lionhearts trailed 3-2 from the first leg in Paris.
Will Cawley won the second-leg opener and victories followed for Calum French and Andras Vadasz before the French fought back to force the decider.
The British Lionhearts won a dramatic tie-breaker to defeat France Fighting Roosters and reach the World Series Boxing semi-finals.