The band will be both playing a gig and attending a film screening and discussion at the University of Otago.
The feminist group has made headlines worldwide with their activism and anti- Vladimir Putin stance, which has seen members jailed in the past.
Pussy Riot has been playing in Auckland, and were to travel to Brazil, but visa issues meant that trip was not possible.
The band will perform at New New New in Dunedin on March 17.
New New New spokeswoman Dallas Synnott said she was contacted by a local musician who had been contacted by Pussy Riot's management.
The band had scheduled a visit to Brazil that was cancelled because of visa problems.
"They decided they love New Zealand.
"They just decided to stay a lot longer and do as many shows as they could while they were here,'' Ms Synnott said.
She described the visit as "very exciting''.
A day before the gig, members of the band will attend a public event at the University of Otago's Department of Media, Film and Communication Studies.
Art, Protest & Power: In Conversation With Pussy Riot will be ''a conversation on the power of art, spectacle and transgression in these chaotic, media-saturated times''.
A film about the band, Act & Punishment will be shown followed by discussion.
Eight members of the band are understood to be attending.
