After three defeats in their last six La Liga matches, Barca went into Wednesday's last-16 home leg with City under pressure.
Their shock loss at Real Valladolid last weekend saw them drop four points behind Real Madrid in the domestic title race, but Messi believes their 2-1 win over City - which completed a 4-1 aggregate victory - shows they can still compete for trophies.
Barca are now in the last eight of the Champions League for the seventh year in a row, and also face arch rivals Real in next month's Copa del Rey final.
Messi's 28th goal of an injury-affected season gave Barca the lead at Camp Nou on Wednesday, and the Argentina captain feels the Spanish champions showed signs of returning to their best.
"We are once again the Barca we want to be," the 26-year-old told the club's official website.
"The performance has nothing to do with the matches we didn't win. We know that we didn’t do the things we were supposed to do.
"La Liga isn't lost because there is still a lot (of games) left."
Barca skipper Xavi, meanwhile, echoed Messi's sentiments about the City game and his side's form.
"I believe we dominated the match, the feeling is good and the objective was to reach the quarter-finals - here we are, (we are) pleased with our victory," he said.
"Our opponents also play, but we tried to keep the ball and play our football.
"The manager (Gerardo Martino) is the one who decides how to play, and we have a big team and are playing well.
"We've recovered our form."
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