The Catalan giants held a 2-0 lead after the first leg at the Etihad Stadium last month so just needed to avoid a two-goal defeat to City which was the case in the Round of 16 second leg as they won 4-1 on aggregate.
Lionel Messi struck his eighth goal of the European campaign with just over 20 minutes remaining to break the hearts of the visitors after they had started the second half in fine fettle.
Although Vincent Kompany pulled one back for City in the final minute of regular time, Dani Alves was on hand to fire home in injury time to ensure the victory, which was in the end harsh on the English side.
An open first half saw both sides produce scoring chances with the best coming when Neymar had the ball in the back of the net but the initial through ball to Jordi Alba from Cesc Fabregas had been adjudged offside.
Late in the first half, Samir Nasri tested Victor Valdes with a thundering attempt before half-time substitute Edin Dzeko, who replaced Sergio Aguero, got in an early shot which the Barcelona gloveman did well to keep out.
Maestro Messi then hit the upright when advancing menacingly on goal before the opener came in the 67th minute when Fabregas' pass saw Joleon Lescott miss the chance to clear allowing Messi in on goal to slot home past Hart with consummate ease.
As City chased the match, Dzeko was clipped from behind by Gerard Pique in the 78th minute but referee Stephanne Lannoy waved away protests for what was a stonewall penalty, infuriating the City players, especially Pablo Zabaleta who received a second yellow card - this time for abuse.
Kompany was then on hand to knock in Dzeko's header from a corner before Andres Iniesta found Alves in close range for the Brazilian to rifle home with the help of a deflection.
Barcelona now make their way into the elite eight of Europe's premier club competition, joining reigning champions Bayern Munich, Atletico Madrid and Paris Saint-Germain, while City at least improved on their last two seasons in the Champions League.
2013/14 UEFA Champions League - Round of 16 second leg
Barcelona (SPN): 2 (Lionel Messi 67', Dani Alves 90+1') Manchester City (ENG): 1 (Vincent Kompany 89', red card: Pablo Zabaleta 78') *Barcelona wins 4-1 on aggregate Camp Nou, Barcelona, Spain - 85,957
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