Napoli had jumped to a 2-0 lead through Matteo Politano and Romelu Lukaku, but needed Alex Meret to deny Milan striker Santi Gimenez to maintain the scoreline.
Luka Jovic then finished impressively to set up a tense finish, which the Azzurri managed to survive.
Afterwards, a relieved Conte stated, “We dominated the first half against a very strong Milan side that started with the ambition of winning the Scudetto and further reinforced the squad in January.
“I had to force the situation with Frank Anguissa, as he returned from international duty with an injury, but we lost McTominay at the last minute, so I had to take the risk. David Neres hadn’t played for a long time either.
“Inevitably, we gradually lost intensity and some fear began to creep in. It’s an important win that keeps us in the slipstream of Inter and also gives us a good cushion from fifth place. Don’t forget, the Champions League places weren’t even our initial ambition, we started out trying to get into Europe.”
He continued: “Winning today was fundamental, our next trip to Bologna is very difficult, but we are still up there. After eight months of hard work, it would be madness not to believe in something extraordinary.
“We know Inter are like a tank, and in my view with all due respect to the others, they are on a different level to the rest of the league. We are still up there and we’ll see what happens. If we recover all the players to full fitness, we can have our say, but only if we get everyone back.”
On winning at the Maradona, Conte admitted he was delighted for the home support.
He concluded: “I settled in perfectly at Napoli, beyond expectations. The first thing I said in pre-season was that it was the first time I’d received so much without giving anything, so I always felt in debt to the Napoli fans.
“I think the lads and I have in part paid them back for that enthusiasm. I just want us all to realise that what we are doing is extraordinary, even miraculous. We only participated in one tournament, playing just two Coppa Italia games, so we need to focus on the next eight games. After that, we can think about the future. That is what I feel I can say.”