The visitors were looking set to lose their first match away from Carrow Road this season after Richie Partridge's stunning effort midway through the first half.
But the Norwich substitute earned his side a draw with his lobbed effort at the death.
Norwich had much the better of the opening exchanges but failed to turn their early dominance into any clear-cut chances.
The home side created the first real opportunity after 14 minutes when Gary McAllister and Lee Mills combined to set up Partridge, but the on-loan Irishman fired just wide from inside the box.
Gary McSheffrey then went close but his effort from the edge of the area was deflected by Darren Kenton and the ball went out for a corner.
The Canaries' best effort early on came from Iwan Roberts who rose to meet Nicky Southall's corner and headed the ball goalwards only to see it cleared off the line by Moroccan midfielder Youssef Safri.
And midway through the half Coventry took the lead through a superb strike from Partridge.
The 22-year-old winger cut inside from the left, knocking the ball through the legs of Southall, before unleashing a shot from 25 yards which whizzed towards goal over the hand of Robert Green and into the top corner of the net.
McSheffrey almost made it two a few moments later but his effort was again deflected wide for a flag-kick.
Norwich were rocked but, to their credit, began to come back into the game as the half approached its close, but Steen Nedergaard wasted their best chance by blasting over the bar from the edge of the box.
McSheffrey could have doubled Coventry's lead early in the second half but his shot was well saved by Green.
David Nielsen carved out an opportunity for himself by skipping past two defenders but his shot fell wide.
Youssef Chippo was gifted an opportunity in the 65th minute when Green misplaced a pass but the Moroccan fired wide from the edge of the box with the whole goal to aim at.
The home side played with five men in midfield as the game drew to a close but still came closest to scoring as John Eustace and David Pipe had efforts saved by Green.
Norwich looked destined to leave Highfield Road with nothing but substitute McVeigh hit an equaliser in the final minute.
The Canaries' top scorer latched on to a mistake by Gary Caldwell before lobbing Fabien Debec from the edge of the box.
|Kickoff 23 October 2002 19:45 at Highfield Road|
|Referee: Pike, M. S.||Attendance: 16,409|
P.Mulryne, 51, unsporting behaviour