The former Liverpool striker fired in two first-half goals as the visitors completely dominated proceedings in the opening 45 minutes, before Aguero pulled one back in a much-improved display from City after the break, while Lionel Messi had a penalty saved in the dying moments.
The hosts??? comeback was halted when Gael Clichy was sent off for a second yellow card, and Barcelona saw out the victory to take a lead back to Camp Nou.
The result is a slight improvement on the sides??? last meeting in Manchester 12 months ago, when Barcelona ran out 2-0 winners on their way to a 4-1 aggregate victory, but Pellegrini???s men know they face an uphill struggle in the second leg of this last-16 tie on March 18.
City were missing Yaya Toure through suspension, but Manuel Pellegrini went with an attacking line-up and named two strikers for the clash.
The tactics didn???t stop Barcelona dominating the early possession and, after giving City a couple of early scares, the visitors took the lead inside 16 minutes as Dani Alves??? cross hit Vincent Kompany and landed for Suarez, who drilled an excellent left-footed strike across Joe Hart and into the corner.
The Spanish side continued to dominate possession and it was one-way traffic ??? a tame Edin Dzeko header which went wide being City???s first real imposition in the Barcelona box.
Things soon went from bad to worse for City as Suarez bagged his second of the night in a move started wonderfully by Messi ??? the Barcelona legend skipping past three men outside the box before laying off to Alba, whose cross was prodded into the corner of the net by the grateful Suarez.
It could have been even more of a mountain to climb for the hosts as a slice of bad fortune nearly cost them, Martin Demichelis??? clearance hitting James Milner and running through for Alves, whose cross-shot flicked behind off the crossbar.
As the half ended City finally registered their first real effort of the match, Sergio Aguero laying off for Nasri, who curled an effort from 20 yards which Ter Stegen parried away without any real difficulty.
But it was a much-changed display from the hosts after the break, and they immediately went on the offensive to try and force their way back into the game. Dzeko wasted a glorious chance as a flicked corner fell into his path but, with plenty of time, he rushed his header and it went straight to Marc-Andre Ter Stegen.
Pellegrini???s men continued to press and make life difficult for Barcelona, and Aguero nearly pulled one back with a curling effort which flew just wide of the upright.
But City???s reward eventually came on 69 minutes when Fernandinho slid the ball towards the area and Silva produced a superb little flick for Aguero, who took the ball away from several defenders before smashing a shot into the top corner.
It was no more than City deserved, but any hopes of completing the comeback were dashed when Clichy ??? already on a yellow for a foul on Ivan Rakitic ??? flew in late on Alves and picked up his second booking to leave the hosts a man down with 17 minutes remaining.
Barcelona had a superb chance to restore their two-goal cushion in stoppage time as Pablo Zabaleta slid in and fouled Messi inside the area, giving the referee no choice but to point to the spot.
The Argentina forward fired his effort towards the top corner, but Hart guessed right and parried it away before Messi missed the rebound with the goal at his mercy, placing his header wide of the opposite post, and City held on to go to Camp Nou one goal behind.