The visitors had the lion's share of possession in the first half, but failed to cause home goalkeeper Joe Hart too many problems.
City soaked up the pressure well and carved out a few opportunities of their own, but Demichelis' foul on Lionel Messi early in the second half represented a turning point.
Barca's control over proceedings inevitably grew with their numerical advantage, and Alves added a second in the closing moments from close range - dealing a huge blow to City's chances of progression.
Alvaro Negredo was City's only out-and-out striker as Manuel Pellegrini looked to bolster his midfield and counter Barcelona's considerable attacking threat, while Gerardo Martino opted for Cesc Fabregas in place of Pedro.
The hosts struggled to get a touch on the ball in the opening stages as their visitors stroked it around with ease but, despite the Spanish champions' early dominance, it was two of the home side's Iberian imports who carved out the first chance of the match - David Silva playing in Negredo, who failed to find the net from a tight angle.
For an encounter between two of the world's most technically gifted sides, the deadlock was almost broken in the scrappiest of fashion in the 25th minute, as Victor Valdes' failure to deal with a Silva free-kick sparked havoc in the Barca penalty area. The goalkeeper eventually got back to his feet to claim the loose ball.
But City's hard work in the first half unravelled nine minutes into the second, as the referee ruled that Demichelis had brought down Messi just inside the area - a challenge that earned the City defender his marching orders.
