Wins in the singles for Alfie Hewett and Gordon Reid meant that the rubber's doubles match was not needed.
Hewett's 7-5 6-1 win over Michael Jeremiasz gave Reid the chance to clinch the title in Antalya, Turkey..
And the Scot sealed the victory over the 2014 champions with a 7-6 (7-5) 6-2 win over Nicolas Peifer.
"Making the transition from juniors and still being able to perform on this stage, it's a very special moment," said 17-year-old Hewett, who had made his senior debut for GB this week.
"I said at the beginning of the week I thought we could win. It has been great to have maintained that belief and to perform to our potential.  It feels unreal."
It was the first time Britain's men had reached the final of the World Group in the 31-year history of the event, which is the wheelchair tennis equivalent of the Davis Cup.
Earlier, Britain's women claimed their third bronze medal since 2011 with a 2-0 win over Germany.
Great Britain's men have won the World Group in the wheelchair tennis World Team Cup for the first time with a 2-0 victory over France in the final.