Danish goalkeeper Casper Ankergren answered Leeds' SOS by filling in for the ill David Lucas and then pulling off a desperate save to deny Walsall striker Jabo Ibehre eight yards out.
It was a rare first-half opportunity for the visitors as Leeds had most of the possession without forcing an early breakthrough.
Leeds finally broke the deadlock in the 46th minute when Walsall failed to clear Delph's corner and Becchio forced the ball over the line for his eighth goal of the campaign.
Delph doubled the advantage in the 65th minute when Andy Robinson rolled a free-kick to him 25 yards out and his left-foot drive rocketed into the top-left corner.
Michael Ricketts wasted a good chance for Walsall against his old club, shooting just wide, before Delph fired in his second and Leeds' third after 87 minutes from 25 yards.