United were five goals up until defender Lubomir Michalik was sent off in injury-time for a second bookable offence and Calvin Zola headed in Billy Jones' cross from the free-kick.
Eugene Bopp then added another consolation goal for the visitors with the aid of a deflection.
Delph showed immense promise against a Crewe side with only one league victory this season and he crowned the performance with a spectacular first senior goal to spark the romp.
The youngster's speculative 30-yard shot struck one post and went in off the other with keeper Steve Collis well beaten in the 26th minute and, 10 minutes later, Collis should have saved Alan Sheehan's long range shot from the left but could only help the ball over his head and in off the bar.
Delph was full of running and repeatedly had Crewe back-pedalling to make it a frustrating return to the ground for former Leeds striker Anthony Elding.
Jonathan Douglas increased the embarrassment for Collis with another goal from well outside the penalty box in the 49th minute.
Substitute Andy Robinson crossed for Jermaine Beckford to head in the fourth on 66 minutes and Robinson fired in the fifth with eight minutes left when he curled the ball into the far corner from 20 yards as Leeds earned their first home league win of the season.