Roberto Mancini's side simply did not look like a side in need of a victory, conceding after just seven minutes when Jason Puncheon slotted home.
Hart was culpable for Southampton's second. Excellent interplay put through Rickie Lambert, whose fierce drive squirmed through the England international's grasp, allowing Steven Davis to bundle home.
Edin Dzeko reduced the deficit as half-time approached, yet Barry passed into his own goal unchallenged to give Pochettino his first win.
Saints dominated proceedings early on and netted a seventh-minute opener. Puncheon burst down the right and played through Jay Rodriguez. Hart managed to deny the winger but the ball fell kindly for Puncheon, who slotted home.
City were rocked by a second Southampton goal in the 22nd minute. Lambert collected the ball and played a neat one-two with Puncheon, before cutting back and firing a shot that Hart let slip through from his midriff, allowing Davis to poke home.
The visitors produced a response six minutes before the break. Bosnian striker Dzeko got himself in the way of a terrible Saints corner, before bursting up the field and tucking home a Pablo Zabaleta cross.
The hosts felt hard done by in stoppage time when Rodriguez was brought to ground in the area by Yaya Toure, only to see referee Martin Atkinson award a corner.
However, Saints did get a third early in the second period as Puncheon could not quite latch onto a Lambert pullback, which Barry inexplicably passed into his own goal.
City were then kept at bay, meaning Manchester United could go 12 points clear at the top should they beat Everton.