Leeds goalkeeper David Lucas was missing through illness so Casper Ankergren deputised, while Michael Ricketts, who was loaned to three clubs during his two seasons at Leeds, captained Walsall.
Leeds enjoyed overwhelming possession with Jermaine Beckford, Neil Kilkenny and Delph all having chances before Walsall almost scored from their first menacing attack in the 22nd minute.
A swift break by Chris Palmer ended with Troy Deeney's right-wing cross finding Jabo Iberhe eight yards out, but Ankergren made a superb blocking save.
A strong wind did its best to blow the ball off course, but patient Leeds refused to abandon their passing game as they sought a way through.
Yet they had to be wary of the swift counter-attack and Iberhe was only a bootlace away from connecting with Richard Taundry's cross into the goalmouth.
When Kilkenny tried the long-range route, Clayton Ince did well to old the Australian international's powerful shot.
Leeds' patience paid off immediately after the interval when Delph's corner was cleared out to him and, when the youngster returned the ball into the danger zone, Becchio forced it home for his eighth goal of the season.
Leeds replaced Kilkenny with Andy Robinson on the hour and his first touch was a free-kick which cleared the bar after hitting the defensive wall.
From the resultant corner, Beckford claimed his shot was handled on the line but to no avail.
However, when Robinson played a free-kick to Delph 25 yards out, the youngster struck an unstoppable left-foot shot into the top left corner of the net in the 65th minute.
Delph added his second three minutes from the end from just outside the penalty box and Leeds managed to keep only their third clean sheet of the season.