Veteran forward Giroud diverted home a wayward shot three minutes after the break at the Stadio Diego Armando Maradona to put Milan two points clear of local rivals Inter Milan.
Champions Inter, who have a game in hand, had claimed the summit on Friday with a 5-0 thumping of Salernitana but now sit second, a point ahead of Napoli.
The Frenchman's eighth league goal of the season was his first away from the San Siro since signing from Chelsea in the summer and came on an eventful night for the 35-year-old.
"I always try to do my job when I'm in the area, to be in the right place at the right time," Giroud told DAZN.
"I don't want to talk about the Scudetto right now but we're hoping to win something this year. We're ambitious and we'll fight to the end."