The 37-year-old Gove, a three-time Nationwide Tour winner who spent the past three seasons on the PGA Tour, matched Norris (65) at 10-under 134 on The Hills course.
Gove played the back nine -- his first nine holes after starting at No. 10 -- in 7-under 29. He had an eagle on the par-4 15th, and four birdies on the final five holes on the side, then got to 10-under overall with birdies on Nos. 3 and 6.
Norris also played the back nine first. He had five straight birdies on Nos. 12-16, and also birdied Nos. 5 and 8.
"That was really, really fun. I haven't done that for a long time," Gove said of his battle with Norris. "In the end we ran out of holes and had to stop."
"We just fed off each other," Norris added.
New Zealand's Steve Alker, the New Zealand PGA winner last week at Clearwater, was a stroke back after a 67 in the event sanctioned by the Australasian and Nationwide tours. Americans Martin Piller (69) and Alex Prugh (71) were two strokes back at 8-under along with Australia's Stephen Dartnall (69).
Prugh was troubled by the gusting and changing winds.
"I definitely thought scoring was out there, I just didn't make as many putts," Prugh said. "Very frustrating. Fortunately I never got really down on myself. I focused real hard to at least birdie my last and get in red figures on the day."
New Zealand amateur Danny Lee, the Johnnie Walker Classic winner last month, missed the cut with rounds of 71 and 77. Also the U.S. Amateur champion, the 18-year-old Lee plans to turn pro after The Masters.
New Zealand's Bob Charles, back in the field after tying for 23rd in December 2007 at The Hills, also missed the cut, following his opening 73 with a 77.
The first left-hander to win a major championship when he took the 1963 British Open, Charles will be 73 on Saturday. He won the New Zealand Open as an amateur in 1954 and added titles as a pro in 1966, 1970 and 1973.