Boone finished 15 of 27 for 181 yards and his second-quarter interception was Duke's first turnover of the season.
Jamison Crowder caught six passes for 119 yards for the Blue Devils, who held just a 437-375 advantage in total yards.
Tanner Lee was 14 of 35 for 160 yards but was intercepted three times — including Fields' late pick-six.
The Blue Devils scored 21 points off Tulane's five turnovers.
The Green Wave's special teams — and by extension, the offense — had major problems on all sorts of fourth downs.
The punt team gave Duke an early safety by snapping the ball through the back of the end zone on a punt, and that was well before the play of the day.
After another poor punt snap was fumbled, Tulane linebacker Rae Juan Marbley scooped up the ball and let it fly. Ajeigbe was there to pick it off and return it 12 yards for the final score.
The Green Wave also missed an extra point, and they went for it — and failed — twice on fourth downs in obvious field-goal range.
Yet Tulane pulled to 26-13 on Thompson's 4-yard scoring run midway through the third, and on its next possession drove to the Duke red zone before a 7-yard pass on fourth-and-8.
Fields then followed two possessions later with his 22-yard interception return for a TD.
Wilson, who rushed for a school-record 245 yards last week against Kansas, finished this one with 70 yards on just seven carries while remaining as the third man in a four-man rotation that also included Snead, starter Shaq Powell and Ajeigbe.
An odd play late in the first quarter gave Duke a 16-7 lead: Tulane long-snapper Matt Marfisi snapped the ball well over punter Peter Picerelli's head and out of the end zone for a safety.
Ross Martin kicked a 31-yard field goal for Duke and kicked six extra points to become the school's career scoring leader with 243 points.