In front of a raucous Manchester crowd, Evans had earlier beaten Arthur Fils 3-6 6-3 6-4 in the opening singles only for Cameron Norrie to lose to Ugo Humbert 7-6(5) 3-6 7-5.
That meant the doubles rubber would decide who would join Australia for the finals in Malaga after they had already secured one of the top two places in Group B.
The French pair strolled through the opening set but Britain levelled the match by winning the second-set tiebreak to set up what turned out to be a deciding set cliffhanger.
Britain looked down and out when Evans double-faulted at 4-5 to trail 0-40 but France were unable to take any of their match points as a fired-up Evans hung on.
France had yet another match point when Skupski served at 5-6 but he stayed ice cool to fire down an ace to keep Britain alive and send the match into another tiebreaker.
Again France had the edge early on and led 4-2 but Britain stormed back to win some mind-boggling net exchanges and they won the match on their second match point to top Group B.
"I don't know how we've won that fight, sheer fight. Incredible effort from Dan to come out and win this match with me. First set it wasn't too good," Skupski said.
"Dan saving three points in the third set. The rest is history and we're off to Malaga."
Evans looked close to self-combustion at times as he whipped the 13,000 crowd into a frenzy and exchanged 'pleasantries' with the French bench.
"A set down, the guys on the bench said get the crowd involved," he said. "Some embarrassing stuff from me. It doesn't matter, we're going to Malaga."
Britain, who last won the Davis Cup in 2015, will now go forward to the eight-nation knockout rounds to be played in the Spanish city from November 21-26.
Italy also clinched their place in the last eight as they beat already-eliminated Sweden to grab runners-up spot behind holders Canada in Group A.
The other nations qualified are Novak Djokovic's Serbia, Canada, Czech Republic, Finland and Netherlands.
