But Zinedine Zidane's men fought back to earn a point as Karim Benzema fired past Yann Sommer in the 87th minute before Casemiro found the net deep in stoppage time to ensure the teams shared the points.
Gladbach now have two points from as many games in the competition and sit two points behind Group B leaders and next opponents Shakhtar.
Thuram was happy with his side's performance against the Spanish champions, but warned that they must be more vigilant in the future.
“We played well on the whole, but we came up against an excellent side,” he told reporters.
“Everyone knows that such a tough opponent doesn’t give in – they fight to the end. We saw that in this match.
“Unfortunately, we conceded two late goals, but I hope we can learn from this and do things better in the next game.”
Thuram needed just two appearances in the Champions League to score his first goals in the tournament, giving him a better record than his father, Lilian.
When told that he had eclipsed his dad, Thuram replied: “I didn’t even know until now that he even scored a goal in the Champions League, but I’m pleased for him!"
But the France international feared his second goal was going to be disallowed and delayed his celebration for a few seconds.
“I thought that I might just have been standing offside. We know what it’s like with VAR, too," he added.
"I didn’t want to celebrate too early in case they intervened, but everything was fine and the referee gave the goal. Then I could celebrate.
“We’ll keep going and work hard to improve. We’ll also obviously try to perform at our best in the upcoming games. After the six group games, we’ll see where we’re at and whether we’ve managed to progress or not.”