The young Frenchman stayed alongside Nibali in the peloton on stage 16 from Carcassonne to Bagneres-de-Luchon and was quickly reeled in when trying to make a break.
Pinot finished over eight minutes behind stage winner Michael Rogers (Tinkoff-Saxo), but was able to put a minute between himself and fourth-placed Jean-Christophe Peraud (AG2R La Mondiale) in the general classification standings.
Nibali expects the new white jersey leader to provide a real threat to his fight for the overall race win and will be monitoring him closely in the remaining five stages.
"I don't underestimate Pinot," said the Italian. "I don't underestimate anyone.
"I've learnt that last year at the Vuelta. When Pinot attacked, I went to catch him straight away. Alejandro Valverde left a gap but I didn't let the gap grow."
Meanwhile, Pinot believes the yellow jersey is out of his reach, but a podium finish is something he would love to achieve.
"This is a great day," said the 24-year-old. "My goal [on Tuesday] was to take the white jersey and save time on my opponents, Alejandro Valverde and Romain Bardet.
"I had very great feelings, my best leg of the Tour, so I had to take this opportunity, I wanted to do something.
"The most important thing is the podium, that would be wonderful.
"But I have to stay calm. Second place seems accessible as Valverde is not far but he is not easy to crack, he never loses a lot of time."