The duo had won their respective singles matches on Friday to put their country in control of their semi-final tie and, despite dropping the first set to the Czech Republic pair, they fought back to record a 6-7 6-4 7-6 6-1 triumph on the clay courts at Roland Garros.
Both teams broke serve once before the opening set reached a tiebreak, and it was the visitors who struck first to edge into an early lead.
However, Tsonga and Gasquet found the early break in the second and that was enough for them to level the match up in front of their home fans in Paris.
The third proved to be a back-and-forth affair, with four breaks of serve leaving another set needing to be decided by a tiebreak, but this time it was France who prevailed to move one set away from a showdown with either Switzerland or Italy.
Losing that set broke the spirit of Berdych and Stepanek and, after they only won nine points on serve throughout the fourth, Tsonga and Gasquet grew in confidence to seal the match in three hours and 35 minutes.