It is the second poll in a week to show Ms Truss ahead of Mr Sunak, although polling has been volatile during the contest so far and could easily change over the remaining seven weeks of the contest.
While the former chancellor topped all five rounds of voting among Tory MPs, it is the party’s wider grassroots membership of about 200,000 people who will decide the winner.
The survey shows the uphill battle faced by Mr Sunak to replace Boris Johnson and secure the top job in UK politics.
He hit back on Thursday evening by saying he was best placed to defeat Labour leader Sir Keir Starmer and win a fifth term for the Tories at the next general election, expected in 2024.
Asked by LBC radio whether he was saying the party would lose under Ms Truss, he said: "That’s what all the evidence that we have today would show, and that’s what our members will need to consider."
The two candidates sparred over their tax plans on Thursday, with Ms Truss promising immediate relief but Mr Sunak trying to show prudence by making more restrained pledges.
Ms Truss said she would undo a rise in national insurance and suspend a green levy to reduce household bills, arguing that tax cuts would stimulate growth and eventually increase revenue.
She promised tax breaks for people who take time out to care for children or elderly relatives, with her team praising carers for easing pressure on the public purse.
Mr Sunak countered that a "huge borrowing spree", with Ms Truss's tax cuts estimated to cost between £30 to £40 billion ($36-48bn), would only make inflation worse.
"I’m worried about the inflation that we’ve got at the moment becoming embedded and lasting far longer," he said.
A Sunak ally, backbench MP, said the former chancellor was "being responsible about the economy, not making promises that he can’t keep," in an apparent swipe at Ms Truss.
"You can’t just have unfunded tax cuts because you have to deal with the debt," he told Sky News. "If you just have unfunded tax cuts, where is that money going to be for vital public services?”.
The exact size of the Conservative membership is unknown, but at the last leadership election in 2019 there were about 160,000 members, and insiders expect it to have grown.
The two politicians will now take part in a six-week tour of campaign events across the UK, before the final result is announced on September 5.