Substitute Tom Pope fired in the 90th minute equaliser after Julien Baudet's header offered the Railwaymen a surprise lifeline.
Swansea looked to be in cruise control at 2-0 following a one-sided first half.
Trinidad & Tobago striker Jason Scotland converted an ninth minute penalty after Danny Woodards' trip on Andy Robinson.
And then a freak own goal from Crewe midfielder Gary Roberts five minutes before the interval appeared to left Swansea home and dry.
Crewe had barely mustered a shot on goal in the first 45 minutes with Swans keeper Dorus de Vries only truly tested when Roberts half a long-range effort at him.
Crewe improved after the restart and Byron Moore warmed up de Vries' gloves with a volley and Ben Rix was denied when the keeper came off his line to save his angled drive.
But Crewe's critical faithful turned on the side and there were chants of 'sack the board' and 'where's the money gone?' as their team appeared to be heading for an inevitable defeat.
The introduction of Pope and Eugen Bopp, who has been strangely marginalised since his summer move to south Cheshire, made a big difference.
Without giant-centre half Dennis Lawrence, who was stretchered off in the first half, Swansea appeared vulnerable from the dead ball.
And Baudet made his presence felt when he powered a downward header up off the crossbar with 20 minutes to go.
Then after Bopp had forced de Vries to palm around his rising drive, the German midfielder produced a deep corner from which Baudet forced a header home from close range with eight minutes left.
Then in the last minute after Bopp had shown notable determination to win possession on the far post and deliver the ball back into the box, Pope crashed the ball high into the net for a dramatic equaliser as the jeers turned to cheers around Gresty Road.