Barcelona, who eliminated Milan last season, surrounded Scots referee Craig Thomson claiming Cristian Zapata had handled the ball before it fell for Boateng who smacked it into the net in the 57th minute.
Barca protested furiously and were unsettled as the match degenerated into a niggly affair, with the visitors infuriating the crowd at one point by keeping possession after a drop ball.
Andres Iniesta was close to equalising with a shot which swerved wide and Barca, who had dominated possession in their familiar style, lost their grip on the game and were sunk by a second goal from Boateng's fellow Ghanaian Sulley Muntari.
"It's very bad [for the return leg]," Barcelona defender Gerard Pique said. "The first half wasn't bad and we controlled the game but when they scored the first goal we lost that control and they were able to take their second chance.
"We have to improve and learn and try to turn things around at the Camp Nou.
"There is no excuse from the pitch or the referee. We'll look forward to the return leg now and try to win," he added.
Even Barcelona top scorer Lionel Messi, who has netted in the last 14 La Liga matches, struggled to make his mark with a muted performance against Milan.
The return leg is on March 12.
Meanwhile, Schalke grabbed a valuable away draw at Galatasaray.
The Turkish side took an early lead through striker Burak Yilmaz.
But shortly before half-time, Schalke levelled when Jermaine Jones scored after a brilliant counter-attack.