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Roberto Mancini's side, who won the title in the dying seconds of last season, displayed the handy knack of scoring late again as Barry's goal broke Reading's resistance and moved City to within three points of United who play at Swansea on Sunday.
City have 39 points from 18 games, followed by four clubs on 30 points after a rainswept opening to the busy holiday programme yielded a mere 10 goals.
A second-half penalty by Mikel Arteta earned Arsenal a 1-0 victory at Wigan Athletic as the Gunners moved up to third place while north London rivals Tottenham Hotspur fell out of the top four after a 0-0 home draw with Stoke City.
Everton climbed to fourth, making light of Marouane Fellaini's absence through suspension, to win 2-1 at West Ham United in game in which both sides ended with 10 men.
West Bromwich Albion also reached the 30-point mark after ending Norwich City's 10-match unbeaten league run with Romelu Lukaku sealing a 2-1 victory 10 minutes from time after Norwich had taken the lead at the Hawthorns.
Chelsea, who have 29 points from two fewer games, can jump back to third place if they beat Aston Villa on Sunday.
There was no pre-Christmas cheer for the bottom four with Reading, Wigan, Queens Park Rangers and Southampton all losing.
QPR's revival under new manager Harry Redknapp came to a halt with a 1-0 defeat at Newcastle United while Sunderland also edged clear of the bottom three after a 1-0 win at Southampton.
City, whose 37-match unbeaten sequence in the league was ended by United earlier this month, were expected to breeze past a Reading side whose defence went missing in a Monday night thrashing by Arsenal but the visitors had not read the script.
Reading were heading for a well-deserved point after restricting City to a handful of chances and even had a reasonable penalty shout waved away but they cracked three minutes into stoppage time.
David Silva crossed into the area as City launched one final assault and Barry, who appeared to have his hand on Reading defender Nicky Shorey, climbed to thump a header into the corner to the relief of the home fans and manager Mancini.
ARSENAL WIN AGAIN
"I hoped, always. It was difficult. We had lots of possession but if you don't score early it is very difficult," Mancini told Sky Sports.
"You have to be patient but if we had scored one goal in the first half it would have changed the game. You need to score quickly in these games and sometimes you take a risk to score. I think we deserved to win.
"Our fans know we try until the last seconds always."
Luckless Reading manager Brian McDermott, whose side will be bottom at Christmas, was angry the goal was allowed to stand.
"I cannot believe that has happened to us. That goal at the end is plain wrong," he said. "(Barry) is on Nicky Shorey's back and it is an absolutely certain foul. That decision killed us."
Arsenal claimed a third consecutive victory and, although they remain 12 points behind United, the storm clouds that were hanging over Arsene Wenger appear to be dispersing.
They were not as impressive as they were in the 5-2 win at Reading on Monday but showed plenty of resilience to grind out three points after Arteta dispatched his spot kick following a clumsy tackle by Jean Beausejour on Theo Walcott on the hour.
Arouna Kone had wasted a glorious chance to give Wigan the lead in the first half and he was denied late on by Arsenal Wojciech Szczesny before Dave Jones thumped a shot narrowly wide of the upright with Szczesny rooted to the spot.
Despite Wigan's late pressure Arsenal held on and will now get a week off after their Boxing Day clash with West Ham United was postponed because of a train strike next Tuesday.
"We had to show nerve and we did it," Wenger said.
"They had a lot of possession in the end but we really wanted it."
Everton trailed to a Carlton Cole goal at Upton Park but levelled through Victor Anichebe after 64 minutes before Steven Pienaar scrambled in the winner.
Cole was sent off for a high tackle on Leighton Baines while Darron Gibson saw red for Everton in the final minute. Both decisions were criticised by their respective managers.
Gareth Bale returned after injury for Tottenham but they missed the chance to end the day in third spot as Stoke proved miserley opposition at White Hart Lane.
Eighth-placed Stoke have kept nine clean sheets in 18 league games and are unbeaten in nine matches.