The host nation needed towin all three rubbers in straight setsto snatch a final spot off the unbeaten Spanish pairing of Maria Jose Martinez Sanchez and Tommy Robredo.
But the Scot responded well, winning the third set 6-2 in just 33 minutes before surviving another tie break towin through in four sets in three hours and 27 minutes.
Tsonga tested Federer in the tournament's opening match on Sunday before losing in threesetsto the defending champion, who is seeking towin a record sixth title.