The host nation needed to win all three rubbers in straight sets to 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 to win through in four sets in three hours and 27 minutes.
The result won't always conform with a sense of how close the matches were: A winner of a five-setter in which he dropped two sets in tiebreakers and won the other threesets 6-0 probably would have a higher dominance ratio than a straight-sets win in which each set was decided by a tiebreaker.
David Estrada scores first MLS hat trick of the season in a 3-1 Seattle win over TFC while Sporting KC sets the pace with three straight wins and will eventually extend that run to seven before losing in April.
Tsonga tested Federer in the tournament's opening match on Sunday before losing in threesets to the defending champion, who is seeking to win a record sixth title.
World No.2 Nadal is struggling with the knee injury he picked up Wednesday's quarterfinal win over Jo-Wilfred Tonga, where he was taken to three hard-fought sets.
The 21-year-old Argentine came into the tournament with a wrist problem, and dropped sets in each of his opening three matches - including a gruelling five-set win over James Blake.