Norwood blasted home from 25 yards after Gary McSheffrey's pass found its way to his feet in the last seconds of a pulsating encounter in south Wales.
Earlier, Cardiff took the lead just after the quarter-hour, with Sky Blues forward Cody McDonald putting through his own net.
Cardiff were thoroughly tested by a Coventry side battling against looming relegation, and were fortunate not to be pegged back when McSheffrey missed a 51st-minute penalty.
McSheffrey then hit a post before Jordan Clarke equalised for Coventry with 20 minutes remaining before Peter Whittingham looked to have won it for Cardiff late in the game only for Norwood to strike at the death.
Cardiff manager Malky Mackay made one change to the side that drew 0-0 with Burnley on Sunday, as Stephen McPhail replaced ankle injury victim Kenny Miller, while Coventry boss Andy Thorn handed recalls to Conor Thomas in midfield and McDonald up front.
Thorn watched his side start brightly and McSheffrey's 20-yard effort flew harmlessly over the Cardiff crossbar.
Aron Gunnarsson fired wide for the hosts at the other end, while Alex Nimely failed to test David Marshall in the Bluebirds' goal as both sides created chances without hitting the target.
And it took an own goal for the home team to edge in front after 18 minutes, as Liam Lawrence's inswinging corner from the left was helped into the back of his own net by McDonald.
McSheffrey shot wide as Coventry looked in vain for an equaliser before the break, and the visitors went even closer when Sammy Clingan's free-kick was almost converted at the back post.
Gunnarsson brought a save out of Coventry stopper Joe Murphy and Whittingham lobbed over the Sky Blues' bar in the dying moments of the opening period, but Cardiff had to make do with a single-goal advantage at the interval.
Coventry started the second half quickly, and McDonald nodded Nimely's centre past a post before the visitors spurned a glorious opportunity to level matters.
Gunnarsson felled Keogh in the area and referee Robert Madley pointed to the spot, only for McSheffrey to fire the spot-kick over the bar.
All the signs pointed to it not being Coventry's, and in particular McSheffrey's, night when the 29-year-old clipped the bar after 65 minutes. But the away side finally managed to restore parity four minutes later.
Clarke seized possession inside the area and produced a fine effort to beat Marshall and find the bottom left-hand corner.
But Cardiff struck back and with seven minutes left Whittingham appeared to have secured all three for the hosts.
The Bluebirds' playmaker-in-chief arrowed an effort into the bottom right-hand corner from 20 yards to cue delirium amongst the home crowd.
But Coventry deserved something to show for their efforts, and with time ticking away Norwood collected from McSheffrey and rifled past Marshall from range to salvage a share of the spoils for the away side.