The Ghanaian international, who netted Milan's second of the evening, believes that it was Allegri's tactics that inspired the first-leg win.
"We listened to our coach, because he has been fantastic in the way he set us up. We were very disciplined on the pitch," BBC Sport quotes him as saying.
"Last week I was telling my wife I've played in the Champions League and never scored so I was very surprised and very happy with the goal and the win for the team.
"Playing in the Nou Camp you can win 5-0 [beforehand] and they can do crazy stuff because it's Barcelona and they have amazing players, but we will try to do well."
Muntari found the net after Kevin-Prince Boateng had put Milan in front, giving them a two-goal advantage to take to Spain.