There was precious little goalmouth action in the first half as Marlon Pack volleyed well over the crossbar, while Adebayo Akinfenwa had a header claimed by Tony Roberts.
The action improved after the break as Akinfenwa lobbed over, while Chris Dunn tipped Damien McCrory's cross-shot around the post.
Johnson then found the top corner of the net to put the Cobblers ahead, but Town were thankful to keeper Dunn for keeping them ahead as he made a stunning save from Peter Gain's volley.
Dagenham kept pushing for an equaliser but leading scorer Paul Benson spurned two late headers. Debutant McCrory almost grabbed a last-minute leveller but his blasted shot was blocked, while Gain had the last chance but his left-footed drive was deflected to safety.