The seven-time European champions came away from Anfield with some lessons to learn after spending the vast majority of their 3-2 defeat to Liverpool on the back foot.
The Italian side returns to the Champions League after a seven-year absence.
The Swedish star, who turns 40 in three weeks time, rolled home the simplest of finishes from Ante Rebic's low cross in the 67th minute.
Giroud joined Milan from Chelsea in the offseason and scored his first two goals for the Rossoneri in Sunday's 4-1 victory over Cagliari 4-1 in Serie A.
He has made a great start in Milan this season after signing from Chelsea, scoring twice in the 4-1 Serie A thumping of Cagliari at the San Siro on Sunday.
AC Milan is aiming to strengthen its presence in the Middle East by signing a partnership with Expo 2020 Dubai.
Olivier Giroud also made his debut for Milan after the veteran forward joined from Chelsea.
AC Milan on Saturday announced the signing of Olivier Jonathan Giroud on a permanent transfer from Chelsea FC.
The 39-year-old pulled up in Milan's 3-0 league win over Juventus on May 9.
The 23-year-old English player arrived last January on loan with the club exercising their option to buy for a reported 28 million euro ($33.4 million).
Two penalties from Franck Kessie helped Milan to a 2-0 win at Atalanta, which had already claimed a top-four spot along with Serie A champion Inter Milan.
The two teams are a point above fifth-place Juventus, which beat Serie A champion Inter Milan 3-2 on Saturday
The 39-year-old recently signed a contract extension until June 2022 and has netted 17 times in all competitions this term.
The match in Turin was a direct fight for a top four finish, with both Milan and Juve locked on the same number of points.
Zlatan Ibrahimovic set up two goals but was also sent off as AC Milan won 3-1 at relegation-threatened Parma on Saturday to keep its small Serie A title hopes alive.