The unstoppable Gareth Bale earlier scored twice in Tottenham Hotspur's 2-1 home victory over Newcastle United while rivals Chelsea and Arsenal also won to add spice to the race for the Champions League spots.
Chelsea stayed third, a point ahead of Spurs and four behind City, after a 4-1 home success over third-bottom Wigan Athletic while 10-man Arsenal beat Sunderland 1-0 away thanks to Santi Cazorla's strike to leapfrog Everton into fifth.
Basement side Queens Park Rangers suffered a big blow in a 4-1 defeat at Swansea City but fourth-bottom Reading failed to increase the six-point gap with a 2-1 loss at Stoke City. Mid-table Norwich City and Fulham managed a goalless stalemate.
Southampton had previously managed two draws and a defeat since Argentine Pochettino replaced the sacked Nigel Adkins in an unpopular move, but their home victory over City was fully deserved even if the visitors gave them a big helping hand.
First Barry lost the ball in midfield after seven minutes and Jason Puncheon stormed forward before coolly finishing off a rebound after Hart's sprawling stop.
Hart, who saved a Ronaldinho penalty during an excellent display for England in a friendly against Brazil on Wednesday, then let a Rickie Lambert shot go through his legs and Steven Davis slid the ball in midway though the first period.
Roberto Mancini had been confidently predicting his side would make up the gap to United, like last season, but he looked stunned in the dugout as City struggled with midfielder Javi Garcia at centre-back after Vincent Kompany was again ruled out.
A rampant break from the edge of their own area gave City hope in the 39th minute, with Edin Dzeko flicking the ball home while falling over after tremendous work by the Bosnian, David Silva and Pablo Zabaleta.
City fans have been used to great comebacks in recent times, not least when they beat QPR 3-2 in stoppage time on the final day last season to clinch the title, but this time there was to be incompetence rather than heroics.
A weak cross came in just after the break and, under no pressure, Barry decided to divert the ball towards Hart only to send it into the corner of the net with the keeper beaten.
"I think that's our best performance... and we got exactly what we deserved," Southampton captain Lambert told ESPN TV.
"We needed that [own goal] in the second half to open up the gap and give us some breathing space."
JENKINSON OFF
Spurs' Wales midfielder Bale netted his fourth and fifth goals in four games for club and country to again inspire the north Londoners in the fourth and final Champions League spot.
Bale struck first with a dipping free-kick after five minutes with James Perch failing to jump in the visitors' wall before Lewis Holtby wasted three chances to increase the lead.
Newcastle's French forward Yoan Gouffran then grabbed his fi
