Hereford lead at the break at Edgar Street, with Aaron Amadi-Holloway converting a Harry Pinchard free-kick to put the Bulls ahead.
Dan Jarvis has hit the post, and debutant Colin Oppong has had a lively first 45 minutes in a Hereford shirt.
Banbury have created opportunities too, with the best from a Josh Smile free-kick which forced a save from Dino Visser.