The Ōtamataha Trust, which represents the interests of Ngāi Tamarāwaho and Ngati Tapu, also put in a claim to be gifted the land in recognition of their mana whenua status and ancestral connection, subject to a lease favouring the foundation.
The foundation planned to develop the property as a reception and education centre for the Elms with facilities to maintain and display its heritage collection.
The trust and foundation had met and agreed to the basic ownership and rental structure.
Discussion of the agreement sparked concerns from some councillors that the transfer would create a precedent for public land on the Te Papa peninsula - from Gate Pā north - to be given back to its original iwi owners, particularly in light of the Anglican Church's formal apology to Tauranga Moana iwi earlier this month over land deals with the Crown 151 years ago.
Councillor John Robson was "concerned about opening the door to that even a little bit", and feared the principles behind the agreement could open the door to recommendations the council transfer all of its land on the peninsula to iwi then lease it back.
"I don't feel like we should be doing that without significant discussion with our community."
Other councillors were adamant the agreement would set no such precedent.
Councillor Larry Baldock said the concerns were "ill-founded" and the two situations should not be compared or used to "torpedo this gesture".
"This is a fantastic step forward in the recognition of the true history of our city."
He said the trust was being treated differently.
"We were not going to consult when it was going to go to The Elms, but now when it's going to Ōtamataha there is a sudden need to consult."
Trust manager Alan Tate said the land transfer was a "one-off issue", not related to the apology or any other land.
He said he was pleased the council's decision would allow the issue to progress.
The trust's chairman, Puhirake Ihaka, and deputy chairman Peri Kohu said they wanted time to consider the decision before commenting.
The Elms Foundation took a neutral stance on the council's decision.
How they voted
To transfer 11 Mission St to the Ōtamataha Trust, to be leased to The Elms Foundation:
For: Terry Molloy, Larry Baldock, Max Mason, Kelvin Clout, Bill Grainger, Steve Morris, Rick Curach.
Against: Leanne Brown, John Robson, Catherine Stewart, Greg Brownless.