The Bosnian should have had a hat-trick after being sent through by Silva in the final minutes of the first half but hesitated and was foiled by Aris goalkeeper Michail Sifakis.
But Toure wrapped up the win to ensure City now travel to Ukraine on 10 March for the first leg of a last-16 tie featuring legendary striker Andriy Shevchenko.
Mancini claimed Balotelli is still not firing at 100% after a recent knee injury, although there were signs of improvement and the Italian did hit the post with one second-half effort.
However, with injuries already ravaging his squad, Mancini can only hope the hip problem that forced Vincent Kompany off before the break proves not to be too serious.
"We won't know what has happened until tomorrow [Friday] when we see what the situation is," explained Mancini. "We have a lot of injuries at the moment, so I hope it is not serious and he will be available for Sunday [the visit of Fulham]."
Aris coach Ioannis Michalitsios was refreshingly honest as he assessed his side's downfall, even though the Greek outfit had gone into the tie level after the first leg.
"We lost to a better team," he said. "We found ourselves two down in the first few minutes, which obviously knocked us back. Given what happened, the eventual outcome could have been a lot worse.
"In order to knock Manchester City out we needed a perfect performance and for them to make a lot of errors. That did not happen. City are a better team than most in Europe and the favourites to win the Europa League this season."
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.