Mathieu lost just three games in the second and third sets in defeating Davydenko, the world's No. 4 player, 2-6, 6-2, 6-1, 7-5 on the indoor red clay court in Moscow.
Russia escaped the day tied with Mikhail Youzhny's stirring 4-hour, 48-minute victory over Richard Gasquet, 6-2, 6-3, 6-7 (8-10), 5-7, 8-6) Youzhny committed 80 unforced errors but also had 13 aces and six service breaks.
The winner of that match will have a 2-1 lead going into Sunday's reverse doubles with Davydenko playing Gasquet and Youzhny meeting Mathieu.
The Russia-France winner moves into the Davis Cup semifinals in September against the winner of this weekend's quarterfinal between Germany and Belgium.