Adel Taarabt marked his Milan debut with a goal to put the Rossoneri ahead, but Gokhan Inler replied immediately.
Higuain then struck twice as Rafael Benitez's men coasted to a deserved victory which re-establishes their three-point advantage over fourth-placed Fiorentina and takes them to within three of second-placed Roma, who face Lazio in a derby on Sunday.
Udinese pulled away from the relegation zone thanks to a 3-0 win over Chievo.
Antonio Di Natale's third goal in his last four games unlocked the game for the Friulian side.
Bruno Fernandes and Emmanuel Badu completed the scoring to take Udinese nine points clear of the drop zone.
In the day's other game, Fiorentina bounced back from two straight defeats to put nine points between themselves and fifth-placed Hellas Verona thanks to a 2-0 win over Atalanta.
A Josip Ilicic free-kick and late Rafal Wolski strike were enough to see off the challenge of Atalanta, who ended the game with 10 men after losing Luca Cigarini to a red card nine minutes from time.