The Tykes took the lead in the 18th minute with a stylish goal from winger Martin Devaney.
He collected a header from Bobby Hassell and, as the City defence parted, Devaney darted unchallenged into the gap and struck a fierce curling 20-yard shot that David Marshall could only divert onto the post at full stretch before it went in.
But the goal sparked Norwich into life and they had a string of efforts that came close to earning them an equaliser before the break.
The leveller eventually came three minutes into the second half when Ched Evans drilled a cool finish through Heinz Muller's legs.
City then took the lead in the 70th minute when Mark Fotheringham crashed in a shot that took a wicked deflection to wrong-foot Muller.
And ten minutes later veteran striker Dion Dublin rolled back the years as he produced a towering header from Fotheringham's corner for the third to send Barnsley to their first home defeat since the opening day of the season.
Barnsley had started brightly with Jamal Campbell-Ryce showing slick control down the right before cutting inside, twisting and turning and lashing a long-range shot just over from the edge of the area.
And when Marciano van Homoet swung in a dangerous cross, Marshall had to stretch to palm the ball away before Campbell-Ryce found space again to drill a low effort across goal.
Kayode Odejayi was inches short of a scoring connection, and when Devaney hit another carbon-copy cross, Norwich were grateful to clear the ball away.
But then came the home side's goal that triggered Norwich's revival.
Barnsley keeper Muller dived low to his right to save after Evans had hit an angled shot and Mo Camara was just short of finishing off the rebound.
Lee Croft sprinted to chase a loose ball and swung in a super cross that brought a diving header from Fotheringham that was just off target.
And another swift break from Norwich saw Evans doing well again to produce a dangerous shot from the right but his effort was a fraction wide.
But after the break Norwich took control after the equaliser as they romped to a comfortable win.