France forward Giroud tapped home his fourth league goal since arriving from Chelsea in the summer in the 14th minute of a largely dreary contest to move Milan three points clear in first place.
Napoli could however reclaim same top spot today with a home win over Bologna.
"I'm very happy, we did well early in the match and afterwards Torino didn't create lots of chances," said Giroud in faltering Italian to DAZN before switching to English.
"We have the three points which is the most important thing. We play every three days and it's very difficult for the body, so everything was not perfect but we are very pleased with the win tonight."
It was a sixth straight Serie A win for Stefano Pioli's side on the day Milan announced they had halved their losses to 96.4 million euros.