A 2-0 defeat on Wednesday meant Gennaro Gattuso lost out to former Milan team-mate Andrea Pirlo in the battle of the coaches.
Goals from Cristiano Ronaldo and Alvaro Morata came either side of Napoli missing a penalty, with Lorenzo Insigne off target from the spot.
Koulibaly, who has been among Europe's most coveted defenders in recent seasons, says Napoli will not be broken by the disappointment of missing out on the trophy.
"A defeat gets even harder to accept on the day after. But it doesn't last long," he wrote on Instagram.
"The desire to fight for new wins is stronger."