Tony Grant's deflected shot ten minutes after the break provided a deserved equaliser for the Clarets after Steve Jones had put Crewe in front in the 17th minute.
But as Crewe manager Dario Gradi was about to make a double substitution the lights went out with 13 minutes left.
It took technicians over 20 minutes to restore power to the pylons but Burnley still lit up the show when play resumed and could have stolen all three points.
Micah Hyde hit the side-netting and Jean Louis Valois pulled an effort wide of the far post.
Previously the Frenchman had been the star of the show for the visitors with some tricky play on the left. John Oster was also a threat on the other flank
Despite having bags of possession Burnley struggled to create clear-cut openings with just Graham Branch left on his own up front. Hyde hit a 20-yarder just over Clayton Ince's bar early on and the Alex keeper had to pull down a shot from Oster.
But a muddle on the halfway line allowed Crewe to break and Jones played a one-two with Luke Varney before the ball ran kindly for him inside the box and he fired a strong rising drive past Brian Jensen into the top corner.
Ince knew little about Valois' thumping volley in the 32nd minute but somehow managed to clear it off his line.
Burnley were still controlling the game, but in a rare attack for the hosts Luke Varney had a neat header from a Kenny Lunt free kick sail just over the bar.
Crewe captain Lunt, who had been taking some stick from the home crowd, livened up after the break hitting a long-range effort just over.
The midfielder's 70th minute free kick was glanced just wide by the head of Steve Foster as Crewe came back into contention following Grant's equaliser.
That had arrived again after some poor defending with loan signing Peter Whittingham and Hyde combining to squeeze the ball through to the former Everton and Manchester City man who profited from a deflection off Lunt as the ball flew into the far corner.