World number five Radwanska had levelled the final when she edged out Cornet in three sets.
France, runners-up at the team format event in 1998 and 2012, got off to a good start when world number 10 Tsonga saw off a spirited challenge from Panfil.
Tsonga took the first set in less than 30 minutes but could not convert any of the seven breakpoints that came his way in the second as Pantil struck back.
He never looked like losing, however, and was largely untroubled in the deciding set.
Radwanska also comfortably won the first set and led 5-3 in the next before Cornet overcame some breathing troubles and claimed the second after 80 minutes of fierce battle.
Radwanska eventually won 6-3 6-7 6-2 against her drained opponent who was given a time violation.
"I was trying to tell the chair umpire not to give me a time violation," Cornet said. "I was like 'I can't breathe'. He didn't see it unfortunately.
"Fortunately I got my breath back a few seconds after and I could continue."
(Reporting by Amlan Chakraborty in New Delhi; editing by Martyn Herman)
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