Giroud tapped in what turned out to be the decisive goal two minutes before the break after a blistering run from Rafael Leao, who was once again key as Milan beat Napoli for the third time this month.
The France striker's 13th goal of the season saved his blushes after he missed a penalty midway through the first half.
Stefano Pioli's side now have the prospect of a local derby in the last four as Inter Milan hold a two-goal lead ahead of their second leg with Benfica on Wednesday night.
"I'm very proud of this team, with that spirit we can dream of anything," said Giroud to Mediaset.
"In the next round there will be another hard match whether it's against Inter or Benfica, but if we keep this spirit we can go anywhere."
After the match Mediaset showed Leao a video of Ruud Gullit setting up a goal with an almost identical run at Napoli in 1988.
"It's the same," said a shocked Leao.
"It's what I try to do, go at players, it's how I can make difference. If I can't score I look for my teammates. I saw Giroud there."
Milan have not contested a Champions League semi-final since they were crowned kings of Europe for the seventh and most recent time in 2007.
Victor Osimhen scored in stoppage time but it was not enough for Napoli who are creaking as they come towards the end of what has been a phenomenal season.
Luciano Spalletti's once freewheeling team are struggling for goals and Khvicha Kvaratskhelia also had a penalty saved with 10 minutes remaining, a miss which condemned his team to defeat.
Napoli are on the verge of their first league title since 1990, but after being placed on the easier side of the draw, Tuesday's exit was an underwhelming end to their European adventure.
