Andrea La Mantia scored the only goal of the game with a diving header at the Stadio Franchi.
“Congratulations to Lecce, because any newly-promoted team that wins away deserves applause," Montella told DAZN.
“We had 19 shots on goal compared to six, we created constant chances, played in attack the whole time without managing to find the net.
“After conceding the goal, we got a little frenetic and that was the only issue in my view. We expected to win considering the performance, the lads gave every last drop of sweat and I am proud they can hold their heads up high."
Franck Ribery was forced off at halftime with an apparent ankle injury.
Added Montella: "I don't know how he's feeling. It was a very tough blow to his ankle, he was upset at not being able to continue the game. I saw the right attitude and desire to win from everyone, so all I can do is thank them. Even a draw would've felt disappointing after that performance, but that's football for you."