The Catalan giants took full advantage of the dismissal of the Argentine to leave Manuel Pellegrini needing a minor miracle at Camp Nou in order to avoid an exit
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 Messi early in the second half represented a turning point.
The defender was dismissed for bringing his fellow Argentine down after he had been played in by a sublime Andres Iniesta pass, and Messi duly stroked home the resulting spot-kick for his seventh goal in four Champions League outings this term.
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 manners 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.
Messi got his first sight of goal on the half-hour mark but scooped a right-footed effort over the crossbar, before Xavi forced Joe Hart into a diving save with a speculative effort from 20 yards.
Negredo headed just wide of the target from a Jesus Navas cross on the stroke of half-time as the sides went in level at the break.
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.
Messi picked himself up to calmly knock the ball straight down the middle past Hart.
City brought on Joleon Lescott as the focus shifted towards damage limitation ahead of the second leg, but could have found themselves two down when Alves drove the ball just wide of Hart's far post.
Silva took the ball down on his chest superbly in the Barcelona area 15 minutes from time, but Valdes got down well to keep out his eventual strike, before Pique had a goal disallowed at the other end after Cesc Fabregas was deemed to be offside.
Alves then stormed into the box in the 90th minute to slot home a goal that puts Barca in the driving seat ahead of a second leg on home soil.