However, Kilmarnock will wonder what might have been if an equalising counter from Steven Naismith had been allowed to stand.
Celtic took control from the outset with the midfield trio of Aiden McGeady, Shaun Maloney and Nakamura keeping Kilmarnock pegged back.
Nakamura was fouled by Steven Naismith in the inside right channel in the seventh minute and he dusted himself down and curled a glorious goal past Alan Combe from 25 yards.
Killie stemmed the tide without worrying the Celtic goal until the final minute of the half when Gary Wales quickly tapped a free-kick to Naismith, who raced through and tucked the ball home only for the linesman to rule offside, a clear and critical error.
Another turning point in the match came around the hour mark. Steven McManus allowed the ball to slip over his foot giving Naismith a clear run on goal. The teenager drew the keeper but slid his shot just wide.
Celtic then swept upfield and Maloney reached the dead-ball line and cut the ball across the face of goal for Hartson to tap home number two.
Killie refused to buckle and Colin Nish got behind the defence and found Wales free at the far post, but David Marshall produced an astounding save to keep the lead intact.
Danny Invincibile and Nish both went close for Killie, before Nakamura danced round the defence to chip home for number three with eight minutes left and Dublin was left alone in the penalty box a minute later to fire home the fourth.
Nish slid home scant consolation for Kilmarnock in the 88th minute.