The visitors kicked off and after a scrappy start Norwich took the lead in the 17th minute when Darren Huckerby smashed the ball inside the near post after the visitors failed to clear McVeigh's left-wing cross.
Four minutes later Huckerby tried to curl a shot past Marcus Hahnemann but the Reading keeper collected the ball safely.
A mistake by Malky Mackay gifted Reading an equaliser in the 24th minute.
The City defender was trying to shepherd the ball out of play for a goal kick but Nicky Forster whipped the ball off his toe and from the left-hand side of the penalty area gave Robert Green no chance with a powerful right-foot angled drive which found the far corner of the net.
Forster looked odds on to put Reading ahead in the 33rd minute but Craig Fleming got in a saving tackle.
In first-half stoppage time Huckerby's 20-yard chip cleared the crossbar by inches.
The home side were put under pressure at the start of the second half. When they finally managed to fashion a goalscoring chance the Reading keeper blocked Phil Mulryne's shot with his legs and was quickly off his line seconds later to deny Huckerby a shooting chance.
In the 68th minute Mulryne's free-kick landed on top of the net and three minutes later Peter Crouch mis-kicked after taking the ball past Hahnemann.
McVeigh hit a long-range effort straight at the keeper before Steve Sidwell limped off to be replaced by Kevin Watson and for Norwich Iwan Roberts came on for Crouch.
McVeigh set up Huckerby with a neat back heel but City's loan signing was unable to find the target.
When it looked as if Norwich would drop their first home points McVeigh unleashed a 25-yard right-foot drive which took a deflection of Adi Williams heel to deceive Hahnemann.
The Reading keeper tried to make amends in stoppage time when his goal-bound header from a stoppage-time corner was cleared by Mackay, who had a good game apart from the one lapse which led to the Reading equaliser.
Norwich: Green, Edworthy, Mackay, Fleming, Shackell, Mulryne, Francis, Holt, McVeigh, Huckerby, Crouch. Subs: Lewis, Briggs, Henderson, Roberts, Easton.
Reading: Hahnemann, Murty, Mackie, Williams, Shorey, Murray, Hughes, Sidwell, Newman, Forster, Goater. Subs: Ashdown, Watson, Harper, Savage, Daley.
Referee: K Hill (Hertfordshire).