The Spanish outfit dominated proceedings in the first period, but stern defending from Milan restricted their ability to create clear chances.
It was Milan that took a shock lead in the second period, however, when Boateng found the bottom corner from close range.
The expected response from Barcelona just did not materialise, with Muntari adding Milan's second 10 minutes from time.
Here, Sports Mole dissects the 90 minutes of action between the two European rivals.
On target: 3
On target: 1
Was the result fair?
When Barcelona lose, there is usually a case to be made that they were unlucky, but not this evening. The fact that they registered just one attempt on target tells you everything that you need to know. Not only were Milan dogged and strong in defence, they had players capable of delivering in the final third. What a second leg it promises to be at the Camp Nou next month.
AC Milan's performance
Where has this team been? On this form, you have to question why they have struggled in Serie A this season. Their league form has taken a turn for the better in recent weeks, which certainly would have boosted their confidence ahead of this match. Many teams try and fail to press Barcelona, but Milan were just sensational throughout the entirety of the match.
Barcelona entered the San Siro as the strong favourites to take a lead into the second leg. However, from the very first moment the tempo of their passing just never came together. Lionel Messi had quite possibly his poorest game of the season, while Andres Iniesta and Xavi also barely figured.
Sports Mole's man of the match
Kevin-Prince Boateng: The former Portsmouth midfielder was just stunning this evening. In terms of winning this accolade, the fact he scored the opening goal is irrelevant. It was his defensive work and intelligence on the ball that epitomised everything good about the Italian outfit.
Barcelona are without question the best passing team on the planet. However, they simply did not get started this evening, with Xavi, Iniesta and Sergio Busquets all lacklustre in possession. It may well have cost them the chance to advance to the quarter-finals.
Craig Thomson had a fairly successful 90 minutes of action this evening. The 40-year-old waited until the 25th minute to award the first yellow card despite a host of challenges, but he was sometimes guilty of not allowing the play to develop. Barcelona were also infuriated by his decision not to award a free kick for handball in the build-up to Milan's first goal.
AC Milan: Milan will play as the away side when they face rivals Inter Milan in the San Siro on Sunday.
Barcelona: The Spanish outfit will attempt to extend their lead at the top of La Liga when they welcome Sevilla to the Camp Nou on Saturday.