The second from bottom club kept former favourite Dean Ashton off the score-sheet on his return but the Norwich striker set up McVeigh's late winner with a charge down the right wing.
He was denied by loan keeper Ross Turnbull on more than one occasion in a first half in which Crewe had the better chances.
As early as the second minute Turnbull had to parry Ashton's angled shot away from goal and came up with a terrific stop to prevent the centre-forward's curling effort from hitting the far corner of the net on the stroke of half-time.
The Middlesbrough youngster, whose loan spell at Crewe might be extended, also made a reaction save to keep out Dickson Etuhu's flick after another former Crewe man, David Wright, had crossed the ball over.
But by then Crewe were ahead and deservedly so as they looked sharp on the counter-attack with Steve Jones in fine form.
It was Jones who crossed from the right in the 15th minute for namesake Billy to thrash the hosts in front at the near post. Robert Green made a super save to keep out a Mike Higdon header and David Vaughan should have done better when he volleyed wide from a juicy position on the edge of the box.
Crewe were to rue missing those chances and, after the introduction of Darren Huckerby on the left, Norwich turned the screw.
Huckerby crossed to the unmarked McVeigh and he stooped to head the equaliser on the edge of the six-yard box in the 63rd minute. Etuhu missed a sitter after he rounded Turnbull and Carl Robinson was put through on goal by Huckerby but failed to appreciate the time he had to beat the keeper and instead squared a pass into no-man's land.
But when Ashton charged clear down the right past former team-mate Richard Walker, Crewe were fatally exposed and McVeigh headed in at the near post from his strike partner's pull-back.