"Now we are among the big guns in Europe," Bordeaux president Nicolas de Tavernost told the French sports daily L'Equipe on Thursday, the day after his side defeated Olympiakos 2-1 for a 3-1 aggregate triumph.
Coach Laurent Blanc agreed with his president but said France needed to cement their position among the elite over a period of time.
"You can think that because of the teams now qualified for the Champions League quarter-finals and those doing well in the Europa League, this proves that French football is in good shape," said Blanc.
"But to affirm this we will have to see how this develops over time to see if this can happen again and to see how regularly French teams advance to the quarter-finals."
There could be more joy on Thursday with Lille starting their Europa League last-16 return leg at Liverpool with a 1-0 advantage and Olympique Marseille playing Benfica at home after drawing 1-1 in the opening match.
Six years ago Lyon were eliminated in the quarter-finals by eventual Champions League winners Porto and Blanc was cautious about Bordeaux's chances in the next round.
FIRST ROUND
"Among the eight quarter-finalists seven want to meet Bordeaux," he told reporters. "Let me remind you our goal was to get through the first round."
No French team has won the Champions League since Marseille defeated AC Milan 1-0 in the 1993 final.
Lyon were given a tough task when they drew Real in the last 16 but they knocked out the nine-times European champions last week, snatching a 1-1 draw in Madrid after winning the first leg 1-0.
France have two teams in the last eight, as many as England, and one more than Germany, Spain, Italy and Russia.
Bordeaux and Lyon have leaned heavily on their world-class performers, with France playmaker Yoann Gourcuff back to his brilliant best for Bordeaux and Argentine striker Lisandro Lopez a huge motivator for Lyon.
"Tonight it's for life and beyond the grave. We cannot leave this stadium without having given everything. Have no regret. Good luck," Lisandro wrote on a board in the Bernabeu dressing room.
Gourcuff's delicate turns and a whipped free kick for a fifth-minute goal reminded everyone why the former AC Milan player has been compared to Zinedine Zidane.
"When he manages to be in charge of the game it feels good for everyone in the team," said Bordeaux striker Marouane Chamakh, who hit the late second goal in Wednesday's 2-1 victory.
