Kay's 12th-minute dismissal changed the course of the game after Barnsley had started strongly.
The Canaries immediately seized control, but had to wait until the second half to really hit their stride, as Darren Huckerby, in particular, tormented the stretched Barnsley defence.
Norwich's Robert Earnshaw fired a shot on target inside the first minute, but it was easily dealt with by Tykes goalkeeper Nick Colgan.
However, Earnshaw did put City ahead from the penalty spot in the 13th minute. His low right-foot shot to the keeper's right sent Colgan the wrong way.
The spot-kick was awarded when Earnshaw collected Huckerby's throughball and appeared to be bundled over from behind by Kay, who was adjudged to be the last defender by referee Iain Williamson who had no hesitation in producing a red card.
As so often happens, the ten men raised the tempo of their game and it was the 27th minute before Carl Robinson doubled the score with his first Norwich goal.
It was a real gem. Lee Croft slid the ball to him from the right wing and Robinson's right-foot shot from some 25 yards found the net via the inside of Colgan's right-hand post.
To their credit Barnsley gave as good as they got for the next 15 minutes and pulled a goal back when Paul Hayes fired in a right-foot shot that took a deflection off Jason Shackell's outstretched leg and beat Paul Gallacher's despairing dive.
Eight minutes into the second half Huckerby picked out Croft who side-stepped his marker, only to drag his shot wide of the target.
However, when Huckerby's low left-wing cross reached Croft at the far side of the six-yard box his right-foot shot zipped beyond Colgan.
Huckerby made it 4-1 in the 64th minute when he cut in from the left wing and let fly with a shot from 18 yards which took a deflection on its way to the far corner of the net.
He then turned provider again five minutes later, when his in-swinging left-wing cross was headed in at the far post by Earnshaw to seal a comprehensive victory.