David Vaughan earned Crewe a winning start to the season with a brilliant late strike.
But the Cheshire side's victory was marred when visiting manager Steve Cotterill made allegations that one of his players had been subjected to racist abuse by a home fan.
Crewe are looking into the complaint, but Cotterill said he would be making a statement to police about the incident after complaining to the hosts' stewards.
It took the gloss off an entertaining win for Dario Gradi's men on the opening day.
Just when it seemed the Railwaymen would have to settle for a point after their first-half ascendancy had been quashed, Vaughan unleashed a wonderful right-footer from 25 yards into the top corner for the winner two minutes from time.
Luke Varney wasted a good early chance for Crewe after he had fended off John McGreal and Wayne Thomas, but after running in on goal he shot straight at Danny Coyne, when it appeared colleague Steve Jones was better placed to finish the chance off.
Defender Darren Moss forced the visiting keeper to tip over with a long-range effort and Varney and Eddie Johnson went close with shots as Crewe called all the shots in the first 45.
They got their reward after McGreal brought down Steve Jones just outside the box and Kenny Lunt pulled the free-kick across the area, where teenager Billy Jones' mis-hit drive found the bottom corner.
Burnley fought back after the break and Micah Hyde's snapshot was well held by home keeper Ben Williams, who made an even better save to prevent James O'Connor's rising drive from finding the corner after Ade Akinbiyi did well to hold off Adie Moses.
But there was no reprieve for the hosts when Thomas headed on Wade Elliott's cross and Noel-Williams sent a header over Williams and into the far corner.
Noel-Williams then came close, hitting the side-netting ripple with an angled drive before being denied by Williams as he looked to have levelled Vaughan's late strike.
The Crewe keeper arced back to tip the striker's deflected shot onto his own bar and rode his luck when the ball bounced back into his hands.