A dominant City cruised into the last four after Silva gave them a third-minute lead and Aguero struck in the 67th minute, with the visitors missing numerous chances to win by a bigger margin.
Pep Guardiola's attacking formation, featuring Silva, Raheem Sterling, Kevin De Bruyne and Leroy Sane behind Aguero, paid instant dividends as City carved open the home team's defence.
Yaya Toure fed Pablo Zabaleta on the right flank with a defence-splitting pass and the Argentine defender drove a perfect ball across the face of goal for Silva to shoot home from close range.
Middlesbrough kept getting undone on either wing as Sane and Silva gave their markers a roasting and the home team's goalkeeper Brad Guzan made a string of good saves.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
Aguero hit the woodwork after another lightning raid down the right but made no mistake when he steered a first-time shot past the helpless Guzan at the near post from a fizzing low cross by Sane.
City threatened to run riot in the closing stages as acres of space opened up, but Guzan spared the home side a heavier defeat by denying Sterling and Silva.
Arsenal host non-league Lincoln City later on Saturday.
(Writing by Zoran Milosavljevic,; Editing by Ed Osmond)
