Marin Čilić will have the first chance to get the one point Croatia needs on Sunday when he takes on most likely Jermey Chardy in the opening match.
Herbert struggled again on his serve as the French looked on the verge of a collapse at the start of the fourth, but he fought through a 13-minute game.
On the other hand, the Croats had to play against 11 break points and they suffered three breaks, overcoming a 3-1 deficit in the third set and giving their best to prolong the match as much as possible.
A second double fault from Dodig in that game sealed the break for the hosts, who held for the set.
Two launch parties were held on the scenario of owners of the court - Herbert and May in each set is made one break, and his serve played two break-point.
After the two pairs shared their service for 3-all in the opening six games, the Frenchmen registered their first break in the seventh before sealing the first set 6-4.
France kept their noses in front, though, and the fans ratcheted up the decibel levels as Pavic served at 4-5. He turned toward the French supporters and gestured as if to ask them to make more noise.
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With Croatia leading the final 2-0, it was a must-win match for France when Nicolas Mahut and Pierre-Hugues Herbert took on Croatia's Ivan Dodig and Mate Pavić.
The match was extremely close, with the French team winning 143 overall points to 139 for the Croatians. Before daybreak the French failed to convert a triple match-point that did not affect their morale.
The victory ensures France live to fight another day, although Croatia will still back themselves to clinch their second Davis Cup crown when Cilic faces Chardy in the first of the reverse singles matches on Sunday.
The second set went underway in a similar manner, as Herbert and Mahut had another break in the ninth game and bagged another 6-4 victory.
Croatia will rely on Cilic and Coric again for another Davis Cup title after 2005.
The decisive point of Croatia, trying to bring the first racket of the world in the doubles Mate Pavic and Ivan Dodig.
From there, Dodig and Pavic found their strength, levelling the set at 3-3 by breaking Herbet and then putting the pressure on Mahut.