With a quarter of an hour to go Jim Brennan's cross struck Alexander on the head and bounced into the net as he lay on the ground.
Early on Norwich hit the woodwork when skipper Andy Hughes dived at Darren Huckerby's cross only to see his glancing header brush the upright.
Cardiff's best effort came early in the second half when Cameron Jerome forced Robert Green into a fine save.
Cardiff manager Dave Jones:
"The only way Norwich were going to score was if we put it into our net and that's exactly what we did.
"It was there for us if we had believed a little bit more in ourselves but we didn't, so it's really just frustrating all round.
"We had enough chances to put that ball in the net but unfortunately we left our shooting boots at home."
Norwich: Green, Colin (Fleming 83), Davenport, Doherty, Drury, Jarrett, Safri (Brennan 45), Hughes (McVeigh 9), Huckerby, Henderson, Ashton.
Subs Not Used: Ward, Thorne.
Booked: Colin, Safri.
Goals: Alexander 77 og.
Cardiff: Alexander, Weston (Ardley 90), Purse, Loovens, Barker, Ledley, Whitley, Cooper (Parry 82), Koumas, Jerome, Ricketts (Lee 47).
Subs Not Used: Margetson, Cox.
Ref: D Drysdale (Lincolnshire).