Second-half goals from Kevin-Prince Boateng and Sulley Muntari gave the Rossoneri an unexpected home victory against last year’s semi-finalists in the competition, and favourites to lift the trophy heading into the first leg.
The vice-president lauded his team’s efficiency and heralded their current state of affairs considering their torrid start to the 2012-13 campaign.
"We did very well and I have to say that our coach was very, very good in my opinion," Galliani told Milan’s official website. "This evening [Wednesday] we played a perfect match. We did well to prevent from getting inside the box.
"We did well against them last year as well though. We can play with Barcelona and that’s what I keep saying. Barcelona have a lot of respect for us and when they say they fear us, they’re not lying.
"In these games, the club history counts as well. If I think of where we were after eight games played this season ... football is beautiful as you never know what can happen, but not even the most optimistic fan would have imagined this."
Galliani believes president Silvio Berlusconi deserves some of the plaudits for the victory, and pleaded with supporters to think back on this triumph when defeats inevitably come around.
"We have to dedicate this win to president Berlusconi as it’s the 27th anniversary of his acquisition of the club and all the fans should know who they have to thank," he continued. "President Berlusconi brought this club to the top of the world.
"The fans shouldn’t forget this, even when we lose to Sampdoria at home, not only when we win against Barcelona. Winning games like this needs great fans, a great stadium, history, tradition.
"I remember the final that Milan lost to Real Madrid in 1958. We have a certain history and other teams look at us with a certain respect. When the stadium is full, it instills fear and puts other teams under pressure."
Although Galliani was reluctant to single out players for praise in the aftermath of the European clash at San Siro, he hailed starlet M’Baye Niang for his impact after coming off the bench.
"We played a perfect match and I don’t want to single out any particular player as they all did well," the Milan chief added.
"Niang showed real personality when he came on. It was a great game, and [we showed] great determination and character."
Barca will be hoping they can overturn the two-goal deficit in the return leg at Camp Nou on March 12.