Pearce, who began the round two strokes behind Australia's Chris Downes, took a four-shot lead to the final hole, but dropped two strokes with a double-bogey 6. He finished at 10-under 278 on the Auckland Golf Club's Middlemore course.
Rumford closed with a 68.
Downes (75) and Australia's Nathan Green (67) tied for third, three strokes back at 7-under.
American Darrell Kestner shot a 73 to tie for fifth at 5-under with New Zealand's Eddie Lee (69) and Australians Scott Gardiner (70) and Wade Ormsby (71).
The 49-year-old Kestner, who failed to qualify for the Champions Tour in November, is the head professional at Deepdale Golf Club in Manhasset, N.Y.
American Michael Pearson (73) was another stroke back in a group that included New Zealand star Michael Campbell (75) and Australia's Wayne Grady (74).