The defending champions were being frustrated by bottom-place Reading until Barry rose over Nicky Shorey to head in David Silva's cross.
"It's more important to score early on in games," City manager Roberto Mancini said. "My heart is not strong for these late goals."
City is three points behind United ahead of the leaders' match at Swansea on Sunday.
"It was difficult clearly, because we had a lot of possession and had the chances but didn't score," Mancini said. "Our support know we try until the last seconds — always."
In the tight tussle for the four Champions League spots, Arsenal climbed to third ahead of Everton, Tottenham and West Bromwich Albion on goal difference.
Mikel Arteta's penalty on the hour clinched Arsenal's 1-0 victory at relegation-threatened Wigan after Jean Beausejour was adjudged to have brought down Theo Walcott.
"It looked very, very soft because it's difficult to tell if there is contact or not," Wigan manager Roberto Martinez said.
But it gave resurgent Arsenal a third consecutive league win for the first time this season.
"We didn't find the same fluency going forward," Arsenal manager Arsene Wenger said. "And it was much more a win because we wanted the result than because we dominated the game."
Everton relied on a fortuitous goal to win 2-1 at West Ham, with Kevin Nolan's clearance from Leon Osman's cross inadvertently coming off Steven Pienaar in the 73rd minute.
Everton's comeback began 10 minutes earlier when Victor Anichebe's header cancelled out a first-half goal from Carlton Cole, who was sent off shortly after the equalizer for a high-footed challenge on Leighton Baines.
Everton had midfielder Darron Gibson sent off in stoppage time, but held on to go fourth.
Tottenham has an inferior goal difference to Everton's and strikers Emmanuel Adebayor and Jermain Defoe were both frustrated against a stubborn Stoke side at White Hart Lane.
But West Brom's recent slump was halted by a 2-1 victory that ended Norwich's 10-match unbeaten run.
Romelu Lukaku headed in the late winner after Zoltan Gera ended West Brom's near six-hour goal drought just before the break to cancel out Robert Snodgrass' opener for Norwich.
While West Brom is sixth, seventh-place Chelsea can return to third with a victory over Aston Villa on Sunday.
In the relegation dogfight, Queens Park Rangers is five points behind Wigan but a point ahead of bottom-place Reading despite Harry Redknapp's unbeaten start as manager ended with a 1-0 loss to Newcastle.
QPR only won its first match of the season last weekend against Fulham, and Redknapp is unhappy with the size of some players' salaries.
"There are a lot of players at this club who earn far too much money for what they are; far, far, far too much money for their ability and what they give to the club," Redknapp said. "I don't really want to see the owners have their pants taken down like they have in the past. There's a lot of agents made money out of them."
Fulham's loss to QPR was followed by a 4-0 drubbing at Liverpool on Saturday, which enabled the Anfield outfit to climb to eighth in the standings.
Stewart Downing scored his first Premier League goal since joining Liverpool before the 2011-12 season.
The winger scored Liverpool's third goal at the start of the second half after setting up captain Steven Gerrard's strike before the break.
Martin Skrtel had put Liverpool ahead after just eight minutes while top-scorer Luis Suarez side-footed home his 14th goal of the season in stoppage time.