The hosts faded after a promising start to the decisive Group E clash, where City midfielder Samir Nasri clattered home a 25-yard strike via the inside of the post with an hour played.
Roma's Kostas Manolas saw his header tipped onto the post by visiting goalkeeper Joe Hart before Pablo Zabaleta struck four minutes from time to ensure the Premier League powerhouse progressed in second place behind group winners Bayern Munich.
"We knew even without some players that City were strong and they proved it," Garcia told Mediaset after the match, before identifying Nasri and Manolas' respective efforts as being key to the outcome.
"It was a very balanced game and you need luck too, so tonight fortune chose its team with those two uprights. I do think we could've scored and done something more in this game.
"Manchester City have more experience and maturity at this level, whereas we are learning and need to improve some elements. We also need some even stronger players if we are to reach teams of that level.
"We should've made more of our opening half-hour, when we had City on the ropes. Similarly, you can't allow a player like Nasri to shoot undisturbed from distance.
"They are not the Premier League holders for nothing. I am really disappointed for the lads, as it was possible to do better and qualify, but it shows us the steps we still need to climb in order to be among Europe's top clubs."
Roma's continental season will continue in the UEFA Europa League in 2015, and Garcia is ready to prioritise Serie A ahead of Europe's second-tier competition.
"If we want to play the Champions League again, then we need a great Serie A campaign," he said.
"We have the Europa League now, which is an international tournament, but the league remains our main objective."
