Home defender Lubo Michalik was sent off in injury time for violent conduct and in his absence Crewe scored twice through Billy Jones and Eugen Bopp.
Delph, who signed the contract 24 hours earlier, produced two moments of magic in as many first half minutes to leave Crewe mesmerised.
First his glorious cross-field pass in the 26th minute released Jermaine Beckford, whose shot was parried by Steve Collis and defender Danny O'Donnell just managed to hook the ball away before Luciano Becchio could apply the finish.
Then Delph took a pass from Neil Kilkenny and let fly from 30 yards with the ball striking the inside of the post and cannoning in off the left upright.
Leeds doubled their advantage in the 36th minute with another long-range effort, this time from Alan Sheehan.
Andrew Hughes played the ball to Sheehan on the left and his shot should have been saved, but Collis could only push the ball over his head and it went in off the bar.
Andy Robinson left the bench in the 65th minute and a minute later the former Swansea winger supplied the cross for Beckford to steal in at the near post and head Leeds' fourth of the match and his own eighth of the season.
Robinson was denied a goal by the bar but in the 82nd minute he got his reward with a glorious drive into the net from the left corner of the 18-yard box.
Substitute Jonny Howson should have made it six but headed over from six yards.
In an eventful finish, Jones drove in the free kick awarded for Michalik's offence and Bopp added Crewe's second with a deflected shot.
Former Leeds striker Anthony Elding wasted two first-half chances for Crewe on his return to the ground and was substituted with 27 minutes left.