City got their noses in front thanks to David Silva, with a shot Jussi Jaaskelainen should have saved, and Barry's superb strike before Ivan Klasnic reduced the deficit.
Another right at the start of the second half from Edin Dzeko restored City's two-goal advantage, and although Kevin Davies netted another for Bolton in the 62nd minute, Roberto Mancini's men held on for the win to move to the top of the fledgling Premier League table.
Barry conceded they were lucky to hang on for the spoils.
"We knew what kind of game it was going to be," he told Sky Sports One. "Kevin Davies caused us a tough game and he's hard to mark.
"But fortunately we got the three points and we're absolutely delighted with that."
Barry also reserved praise for goalscorer Silva, who once again proved his worth for City with the opener.
"Silva has started well this season," Barry continued. "He played fantastic for us last year and has been fantastic again in the first couple of games this year as well.
"The manager said in the week we need to score 10 to 15 more goals than last season and we've started well with seven goals in two games. But now we need to continue that."
Mancini was happy with his team's performance overall but warned they will need to tighten up in defence if they are to challenge for the title.
"We played very well," he said. "But we suffered until the last second because Bolton are a good team who will always try to score.
"We can't concede two goals like today. We need to pay more attention. If we want a good championship we should pay more attention in these sorts of situation.
"I want us to score 15 goals more than last year but we can't concede more goals than last year."
The Italian also revealed he is hoping to tie up a deal for Arsenal's Samir Nasri in the near future.
"I hope we can have the player in the next days," said Mancini.
"I don't know what's the problem. I hope this week we can have Nasri, but don't know when. We have 10 days to the end (of the transfer window)."
Bolton boss Owen Coyle is convinced his side can recover quickly from the loss after giving a good account of themselves against the Premier League's big spenders.
He said: "We gifted them a couple of goals today and that leaves a bad taste in the mouth.
"But we'll pick ourselves up. Our strength is always as a group and we showed that today when we picked ourselves up from two down and then 3-1 down.
"We've shown today whoever we play it's certainly going to be a tough fixture for them. On the whole I thought it was a big performance from our group of players."
Source: Team Talk
Source: Team Talk