His round was marred by a bogey at the 18th hole as he missed a regulation putt for par.
That error gave Brown and Lee a sniff - in fact, Brown could have held the lead outright heading into today had he not produced a double-bogey at the par-five 17th.
Brown signed for a six-under 66 but he put his tee shot into the water on 17, which proved costly.
The 39-year-old answered back with a birdie on 18, which capped an impressive round.
"I just got on a roll where I could see the line and all of a sudden putting became a lot easier, which is nice," Brown said.
No Kiwi has won the New Zealand Open since Mahal Pearce triumphed at Middlemore in 2003 and Brown said the locals were desperate to break the drought.
"We have all talked about it all week," Brown said. "We keep getting reminded about it; it's about time one of us did it."
Lee, who shot a two-under 70 in perfect conditions yesterday, is another one of those Kiwis who could get the job done today.
"I am struggling with my ball striking and I would love to go out there and tear it apart," the 40-year-old said. "I feel like a low number is there for me."
Of the other Kiwis in the field, Craig Palmer, Kieran Muir and Gareth Paddison are among a group of four players in a share of fifth at nine-under, while Ryan Fox and David Klein are part of a seven-way tie for ninth at eight-under.
American Rocco Mediate missed the cut by two shots on Friday but made the cut of the pro-am only to withdraw citing "aches and pains" despite him and playing partner Mark Richardson being well-placed in the pro-am standings. Mediate can't have been too sore because he was due to open the fourth round today by playing 18 holes alongside Prime Minister John Key and former Open champion Bob Charles.