The British duo came back from a set down to beat French pair Nicolas Mahut and Edouard Roger-Vasselin 1-6, 7-6 (7-4), 7-6 (8-6) in front of 13,000 fans in Manchester as nerves faltered on both sides of the net.
In a contest of almost three hours, France came close to sealing their passage and will look back at four failed match points and a series of unforced errors.
Britain join Italy, Serbia, Canada, Czech Republic, Netherlands, Finland and Australia in the finals in Malaga in November.
In the opening rubber, Evans fought back from a set down to see off French teenager Arthur Fils 3-6, 6-3, 6-4 and take a 1-0 lead for Britain.
But Cameron Norrie was beaten by Ugo Humbert, leaving the contest level with everything depending on a tense decider in the doubles rubber.
Earlier, Italy became the seventh team to qualify for the final eight with singles wins from Matteo Arnaldi and Lorenzo Sonego giving them an unassailable 2-0 lead against Sweden in Bologna.