They were asked whether they would accept moving from a final salary pension to a less generous scheme.
Members of three unions at sites in Wales, Scotland, South Yorkshire and Teesside all supported the new proposals.
Tata’s offer included a £1bn investment commitment at Port Talbot and no compulsory job losses.
The three unions at the Indian company’s UK plants had recommended that their members, including 6,300 Tata workers in Wales, accepted the deal.
“This result provides a mandate from our members to move forward in our discussions with Tata and find a sustainable solution for the British Steel Pension Scheme,” Roy Rickhuss, the general secretary of Community union, said.
Its members backed the new deal by 72% in favour compared to 28% against, with a turnout on the vote of more than 70%.
Tony Brady, Unite’s national officer, said Tata must now honour its commitments on investment.
“Nothing less would be a betrayal and add to the deep mistrust that steelworkers now have for the company,” he said.
His members supported the pension offer with 76% voting in favour, while GMB union steelworkers also voted in favour by 74%.
“Now that steelworkers have done their bit, it is time for the government to step up and do theirs,” David Hulse from the GMB added.
Under the proposed changes, the British Steel Pension Scheme will close to future accrual, replaced with a defined contribution scheme with maximum contributions of 10% from Tata and 6% from workers.
It also emerged a one-off pension contribution of up to £10,000 could be made to Tata workers in their 50s who plan to retire early.
It is hoped the vote will now bring an end to almost a year of uncertainty for thousands of workers who faced losing their jobs when Tata’s UK business was put up for sale in March 2016.
Wales’ First Minister Carwyn Jones said the vote was a “significant step forward in securing a sustainable future for Tata’s UK business” after what had been “a difficult decision for the trade unions and their members”.
He added: “We continue to work closely with Tata to implement our significant financial support package in order to maintain steel jobs and production at all the Tata sites in Wales.
“We also repeat our calls for the UK government to urgently tackle the UK’s high energy costs and continue to push for greater recognition of the steel industry in their new Industrial Strategy.”
Aberavon MP Stephen Kinnock said: “Today marks the beginning of a new chapter for the British steel industry.
“This is not just about a job or an industry but about a way of life.”
He also called on Tata to uphold its pledge to retain the two blast furnaces in Port Talbot.
Plaid Cymru South Wales West AM Bethan Jenkins said the workforce had “made its sacrifice for future workers”, calling on Welsh and UK ministers to “demonstrate their commitment to workers and the industry”.
“Such a demonstration should include action on energy costs, and investment in a local power plant and a Steel Research and Development Centre,” she said.
A UK government spokesman said the “positive step” was “testament to the commitment of its workforce that they are willing to work so constructively with the owners to secure the future of the plant”.