Lionel Messi opened the scoring from the spot shortly after half-time of the first leg of the last 16 tie after Martin Demichelis was sent off for bringing him down just inside the penalty area.
Daniel Alves then doubled the advantage at the death, putting Barca firmly in the driving seat ahead of the second leg at Camp Nou in three weeks' time.
But head coach Martino believes his players have no time to take their eye off the ball as they continue to battle for silverware on three fronts.
"For 10 minutes during the first half we lost control a little but in the rest of the game we did well," he said.
"We have to remain calm and continue along the same path because there is no time to relax as we must now start thinking about our next game in La Liga.
"We have clear objectives in each tournament.
"We can play better or we can play worse than tonight but in no way are we going to relax."
Midfielder Cesc Fabregeas, meanwhile, believes Barca silenced their critics with victory at the Etihad Stadium.
"We had a great result tonight," he told ITV. "The fans put so much pressure on and that's what makes this so win special. We had a point to prove, and we did it very well.
"Some people were talking a bit too much before the game and now they have to shut up for a few days.
"You never know in football, but in three weeks we will have to play very, very well."
