Veteran forward Giroud took Mike Maignan's place in the Milan goal after his Les Bleus teammate was sent off in the seventh minute of injury time and did enough to ensure the away side won 1-0 and moved two points ahead of local rivals Inter Milan.
"Last night Olivier Giroud became part of Milan history by courageously defending the Milan goal in the final minutes of our match with Genoa," Milan said in a statement.
"The club has decided to honour his performance in our last line of defence by putting him in the list of goalkeepers."
Milan also said that fans could buy a replica goalkeepers' jersey with "Giroud 9" on the back as an homage to an incredible ending to Saturday's tight victory.
Giroud can be seen on Milan's website smiling in the goalkeeping kit -- which on Saturday night he said he would frame -- above a perfect record of zero goals conceded in his seven league appearances this season.
He was brought on as a substitute midway through the second half at the Stadio Luigi Ferraris to help Milan break down a stubborn Genoa.
But instead Giroud was key to holding off a stoppage time siege from Genoa after Christian Pulisic put Milan ahead in the 87th minute.
Giroud could only watch on helplessly as Albert Gudmundsson clipped a free-kick off the crossbar right after Maignan was dismissed and the 37-year-old pulled on the goalkeeper's jersey and gloves.
And he capped an incredible ending to a match, in which Genoa's goalkeeper Josep Martinez was also sent off, by rushing out of his goal to twice deny Genoa and ensure Milan took the three points back to the San Siro.