Sixty-one percent of nearly 3,000 responses agreed to a change of name, with nearly three-quarters backing Welsh Parliament as the best option.
A law to change the name is expected to pass before the 2021 election.
Presiding Officer Elin Jones hoped people would "more fully understand the powers of the assembly and the role it plays in their lives".
But Neil Hamilton, UKIP AM, expressed concerns over potential costs.
The assembly was given the power to rename itself under the 2017 Wales Act. Assembly members had unanimously backed the idea last July after which it was put out to public consultation.
Publishing the responses on Tuesday, Ms Jones said: "Our role today is as a full parliamentary body, with the power to pass laws and agree taxes, and we must continue to work hard to inspire the confidence, trust and pride in the people we serve."
Speaking to BBC Wales, Ms Jones said the Welsh Parliament name will "reflect directly the will of the people of Wales in giving us the mandate to have the powers to create legislation on their behalf".
"We are a parliament in all but name," she said.
Sixty percent of people responding to the consultation said they did not agree that the role of the assembly was well understood.
Just over a fifth of respondents - 22.6% - who responded to the survey said they strongly disagreed or disagreed with the assembly's name changing.
Legislation on assembly reforms - including the change of name - will be published by the Assembly Commission in 2018.
They will include rules on disqualification of assembly members and the design of the committee system.
The commission proposes that assembly members will in future be known as Welsh Parliament Members (WPMs), in keeping with the current title of Assembly Members.
The option was the second least-favoured option according to a summary of the survey conducted for the consultation - with 11.1% of respondents preferring it over other suggestions.
Member of the Welsh Parliament was preferred by 30% while 28.4% liked Member of the Senedd.
Tory AM Suzy Davies, Assembly Commissioner with responsibility for budget and governance, said the assembly would not "waste resources by rushing to make the change", noting that some people responding to the consultation were worried about the cost.
She said until legislation is passed the institution will continue to be known by its current name in order to avoid confusion and to minimise cost.
There have long been calls to call the assembly a Welsh Parliament, with Welsh Tory leader Andrew RT Davies having made the suggestion in 2012.
On Tuesday, Mr Davies said: "The assembly has clearly evolved into a parliament, and it's only right that this is now reflected in its name."
First Minister Carwyn Jones said: "The name Welsh Parliament / Senedd Wales will help to enhance understanding of the assembly and reflect its status as a national parliament, on a par with the other legislatures in the UK."
Plaid Cymru leader Leanne Wood also welcomed the name change, saying the assembly's stature and confidence had evolved over time.
Concern was expressed by UKIP group leader Neil Hamilton, who said: "While the Welsh Assembly has changed as a result of tax devolution, UKIP Wales is concerned about the potential costs of a symbolic name change."
However, when asked to clarify the party's stance on the name change, a UKIP spokesman said it was not "totally opposed" to it and said the concerns surrounded how much it would cost.
An assembly source said Caroline Jones, the party's commissioner, had agreed to the move as part of the assembly commission.
Assembly documents suggested costs may range from between Â£40,000 and Â£150,000.
But Heledd Gwyndaf, chair of Welsh language campaign group Cymdeithas yr Iaith Gymraeg, said the assembly would miss "an opportunity to normalise the use of Welsh" if it it did not choose "Senedd" as the only name.
The National Assembly for Wales is set to be renamed the Welsh Parliament after it was backed in a consultation.