Giroud struck twice in the 75th and 78th minutes after Ivan Perisic had given Simone Inzaghi’s men a first half lead.
AC Milan went into the game four points behind their city rivals but the 2-1 victory has taken them within a point of the leaders, who have played a game less.
"As I said before the game, this was a special fixture. You win the derby, we were struggling in the first half, but after the break for 45 minutes we got back into it," Giroud told DAZN.
"I was hoping for one or two balls into the box, the first was a bit lucky, but the second was a fine [Davide] Calabria pass.
"I am so happy for my fans, I am proud of this team that never gave up.
"We score goals in the box, so it's true I was a bit isolated in the first half, but the whole team didn't play well and we knew that.
"We proved our great team spirit after the break and we are back!
"There's a long road ahead of us, we are still in the running, but there are many games to be played and Inter remain the leaders.
"So let’s win our matches and then see what Inter do."
