Prior to that, veteran Sri Lanka batsman Tillakaratne Dilshan went for a duck as the Falcons got their innings off to the worst possible start, before Chesney Hughes (5) and Wayne Madsen (0) also went to leave Derbyshire stranded on 38-3.
Durtson and Elstone produced a steady partnership, but the latter was eventually caught by Josh Cobb. Wickets continued to fall around the skipper, as Billy Godleman (2), Shiv Thakor (15) and Thomas Poynton (16) all fell.
Despite losing South Africa international Richard Levi (5) in the first over of their reply, David Wiley's impressive half-century helped keep the Steelbacks on course, before he was sent packing by Tony Palladino for 60.
Cobb went for 11 off the bowling of Wayne White, yet Northants continued to close in on their victory target, which they soon reached when Alex Wakely and Steven Crook provided an unbroken stand of 42 to get them home and dry, with 13 balls spare.