United now will visit Millwall in the play-off semi-final first leg on Saturday, May 9, kick-off 12.45pm, with the home leg on Thursday, May 14 - but the Cobblers are sent back to League Two after three seasons in the third tier.
Town defended stubbornly until the 44th minute when Ben Parker crossed from the left, keeper Chris Dunn stayed rooted to his line and Becchio headed in from six yards for his 18th goal of the season.
Knowing their League One survival was in danger, Town showed more urgency after the break, with Luke Guttridge and Leon Constantine shooting close before Leeds pounced again on the hour.
Snodgrass picked out Beckford with a neat ball into the 18-yard box and Leeds' top scorer hit his 34th goal of the season into the bottom left corner.
And Snodgrass then struck in injury time to volley Leeds' third from Robinson's pass.