The 2015 U.S. Open champions Mahut and Herbert are the world's top-ranked pair.
Richard Gasquet, who won the first semi-final point for France on Friday, now takes on Cilic in Sunday's reverse singles while Lucas Pouille plays Borna Coric. "We have to hope, hope until the end and believe that we can win," said France captain Yannick Noah who is without the injured Gael Monfils and Jo-Wilfried Tsonga this weekend.
Mahut and Herbert wasted three consecutive break points in the opening set before the Croats won the tiebreak when the latter sent a forehand long.
The French pair bounced back in the second set thanks to a break in the 11th game but Dodig and Cilic continued to look more solid and accurate. The visitors saved a break point in the 12th game of the third set only to then lose another tiebreak.
It was one-way traffic from then on as the Croatian duo broke for a 2-1 lead in the fourth set and wrapped up victory on Herbert's serve, the Frenchman burying a volley into the net.
The winners will take on Britain or Argentina. The South Americans lead 2-1 in Glasgow.
(Reporting by Julien Pretot; Editing by Tony Jimenez)
