Archer, who is black, was abused as he left the field after New Zealand clinched victory by an innings and 65 runs, the governing body said in a statement. It was unable to locate the person who hurled racial abuse and will be making further inquiries on Tuesday.
NZC said it was an "unacceptable experience" for Archer, and promised "increased vigilance in the matter" for the second test in Hamilton.
The England and Wales Cricket Board said it was also looking into the matter.
Archer tweeted about the incident, saying it was a "bit disturbing hearing racial insults today whilst battling to help save my team."
NZC said on Tuesday it will search CCTV footage from the Bay Oval in an effort to identify the spectator responsible for the racist comments. Security staff at the venue, which was hosting a test match for the first time, had not been immediately able to find the person.
NZC said it has zero tolerance towards abusive or offensive language and will refer any developments to police.
Archer was playing a test outside England for the first time. He took 1-107 in New Zealand's first innings of 615-9 and made a career-best 30 in England's second innings of 197.
The New Zealand team later tweeted support for Archer: "We are shocked and disappointed to hear of the verbal abuse Jofra Archer received after the test today. (England) might be our rivals but they're also our friends and racist abuse is never okay."
New Zealand captain Kane Williamson said he would seek out Archer ahead of the second test, which begins in Hamilton on Friday, to apologize in person.
Williamson, who lives near the Bay Oval, said he wasn't aware of the racial abuse during the match and was "100 percent" shocked to learn of it later.
"It's a horrific thing," Williamson told the Stuff news website. "In a country and a setting where it is very much multi-cultural it's something we need to put to bed quickly and hope nothing like that ever happens again. It certainly won't if there's any influence we can have on it."
The ECB said that "whilst this is a relatively isolated incident there is absolutely no place for anti-social or racist behavior within the game and it is vitally important that all spectators feel able to come forward to report such behavior and feel safe in doing so."