The Algeria wing-back made his debut for the Rossoneri in Thursday's 3-1 Coppa Italia victory over Lazio following his switch from Lecce.
Mesbah played from the start as Milan ensured a place in the last four of the competition, where they will meet Juventus, and he has now turned his attention to the league.
"I was very excited," the 27-year-old midfielder said to www.milannews.it.
"To play your first game for Milan at the San Siro is important.
"I'm very happy with my performance and with the result.
"Now we have another game on Sunday and we must prepare as best as possible.
"I know I have to continue to work hard and earn a regular place in the squad."
Milan will be looking to cap off a great week by taking maximum points.
The Rossoneri bounced back from their 1-0 defeat to Inter Milan with a 3-0 triumph at Novara last weekend.
That result kept Milan second in the standings, one point behind leaders Juventus at the halfway stage of the campaign.
Coach Massimiliano Allegri, who led Cagliari for two seasons before joining Milan in 2010, is still without the services of injured midfielders Alberto Aquilani, Gennaro Gattuso and Kevin-Prince Boateng, and forward Alexandre Pato.
Despite Milan not being at full strength, Cagliari coach Davide Ballardini knows his side will face a stern test this weekend.
"We will have to play a perfect game against Milan if we are to get a result," he said.
Ballardini will be hoping the arrival of striker Mauricio Pinilla will inspire his team in front of goal.
The Chilean forward joined Cagliari on loan from Palermo this week.
The Sardinians have mustered just one goal in their last two league games but head into tomorrow's game riding a three-match unbeaten run.
Last weekend's goalless draw against Fiorentina left Cagliari in 11th place with 23 points.