Rafael Benitez's side came into the game without a win in three Serie A games and on the back of a 3-2 Coppa Italia semi-final first leg defeat against Roma in midweek.
Napoli's lack of form was not evident as they took the game to Milan in the opening stages, but Adel Taarabt gave Clarence Seedorf's charges the lead with a well-taken debut goal after only seven minutes.
Gokhan Inler equalised three minutes later with a deflected left-foot strike and Higuain scored twice in the second half to give his side a deserved win.
The Naples outfit are now three points behind second-placed Roma - who have two games in hand - and the same margin clear of Fiorentina, who beat Atalanta 2-0 earlier in the day and occupy fourth spot.
Higuain, who has now scored 17 goals in all competitions in his first season at the club, was impressed by the manner in which Napoli recovered from their early setback.
"In truth I was calm, as the team has to do well and if I don't score, someone else will. I am happy for the result," he told Sky Sport Italia.
"The game started badly for us, going 1-0 down straight away. The team reacted in a strong and important way.
"I am convinced we are on the right track.
"We had some good performances recently and lacked good fortune, but always showed character.
"I am happy we got the win against a great side."
