Writing in the Daily Telegraph, Sir Robert said Ms Truss was the “right person to take the country forward”.
The Welsh Secretary had previously backed former chancellor Mr Sunak, citing his experience and judgement.
Mr Sunak won every vote among Tory MPs, but Ms Truss is now consistently tipped to win the backing of party members.
The pair are currently travelling around the country taking part in a series of hustings
– events in which they answer questions and put forward their proposals in front of invited party members.
– either postal or online
– will be cast over the coming weeks and a result will be announced on 5 September.
As the new Conservative Party leader, the winner will also succeed Boris Johnson as prime minister on that date.
In his article, Sir Robert says he had initially backed Mr Sunak because he felt that, during the parliamentary rounds where MPs decided on the final two candidates, he was “embodying what we needed.”
“As the campaign has moved on, and as I have listened carefully to both candidates, I have thought deeply about the issues that move me and what I want to see the next prime minister doing,” he said.
“Changing your mind on an issue like this is not an easy thing to do, but I have decided that Liz Truss is the right person to take the country forward.”
Sir Robert has previously described himself as a “strong Remainer”
“One-Nation, moderate Conservative”. Before becoming secretary of state for Wales, he served as justice secretary and prisons minister.
He said he had switched to Ms Truss in part because she had pledged to address his concerns about the Bill of Rights, a proposed piece of legislation intended to replace the Human Rights Act.
Sir Robert said he also wanted to see the next prime minister prioritise growth, saying it would be necessary if the government was to make a “real difference to people”.
“We need growth and investment, and we need to learn to love business again,” he wrote. “That is… the Conservative way of boosting household finances and getting the tax revenues we need to fund public services,” he writes.
“[Ms Truss’s] plans give us our best shot at reaching our potential with the high-growth, high-productivity economy that we need,” he said.
The state of the UK economy, along with the current levels of tax and the rate of inflation, have been central issues in the leadership campaign.
Ms Truss has pledged immediate tax cuts while Mr Sunak has said
They should be left until inflation has been brought under control, although he has pledged to scrap VAT on energy and cut income tax over the next parliament.
Sir Robert also emphasised the need for “party unity” whichever candidate becomes the next Tory leader.
“We cannot expect to have the necessary strength in government if our party is torn in two by the end of the summer,” he writes.