A 31st-minute goal from Ivan Rakitic secured Barca a 1-0 victory over City on Wednesday, completing a 3-1 aggregate win in the last 16 tie.
Goalkeeper Marc-Andre ter Stegen saved a Sergio Aguero penalty in the 78th minute at Camp Nou, giving Manuel Pellegrini's men no way back into the match.
A top-of-the-table La Liga clash against Real Madrid follows for Barca, and Luis Enrique is happy with the form of his team heading into that encounter.
"We expected the City we saw," he was quoted as saying by UEFA's official website. "It was a game with a lot of chances, and we achieved our objective, which was to win the game and get through to the quarters.
"We had some good chances, and go into Sunday's game with Real Madrid in good shape."
Lionel Messi was in fine form throughout the second leg, and his chipped pass to the back post set up Rakitic for the game's only goal - with Joe Hart performing heroics in the Manchester City goal to thwart them from then on.
The Croatia international lauded his team for their performance.
"I think the side did a great job. You saw a great Barcelona performance. We don't want to stop, and now comes the best part of the season," Rakitic said.
"Joe Hart had a great game, but we were good too and we deserved to win. Now we have to enjoy our win and rest well."
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