The poll by Opinium, based on a sample of 1,032 voters, found 39 per cent believed Mr Sunak had performed best compared to 38 per cent for Ms Truss.
The pair sought to land knockout blows in criticism of each other's economic plans during the hour-long TV appearance in Stoke, central England.
Mr Sunak hammered the point there was “nothing Conservative” about Ms Truss’s approach and it would give the party “absolutely no chance” of winning the next election.
Ms Truss, the Foreign Secretary, suggested her rival would lead the country into a recession.
She said she would put an economic growth plan in place “immediately” if she became prime minister, along with a temporary moratorium on the green energy levy.
The increase in national insurance would also be reversed, Ms Truss said.
Ms Truss said her plans would see the government start paying down its debt in three years. “This chancellor has raised taxes to the highest rate in 70 years and we’re now predicted a recession," she said. “The truth is in the figures.”
Mr Sunak said he would like to make sure that his government “always” had policies in place to support people through the crisis in the cost of living.
“You’ve promised over £40 billion [$48bn] of unfunded tax cuts — £40bn more borrowing," Mr Sunak said.
“That is the country’s credit card and it’s our children and grandchildren, everyone here’s kids will pick up the tab for that.
“There’s nothing Conservative about it.”
Ms Truss said no other country was putting up taxes at the moment, something that could tip the UK economy over the recessionary cliff.
"The OECD [Organisation for Economic Co-operation and Development] has described Rishi’s policies as contractionary," she said.
“What does contractionary mean? It means it will lead to a recession. We know what happens when there’s a recession.
Mr Sunak spoke over her, saying inflation was a problem in the 1980s and it is a “problem we have now”.
"We need to get a grip on inflation," he said. "If we don’t do that now, it’s going to cost all of you and everybody watching at home far more in the long run.
“Liz, your plans, your own economic adviser has said that will lead to mortgage interest rates going up to 7 per cent.
"Can you imagine what that’s going to do for everyone here and everyone watching? That’s thousands of pounds on their mortgage bill.
“It’s going to tip millions of people into misery and it’s going to mean we have absolutely no chance of winning the next election either.”
Former Cabinet minister David Davis, a supporter of Mr Sunak, said the Truss policy Liz Truss would push the base interest rate to 7 per cent, which he says will result in mortgage rates of up to 10 per cent and destroy the party's appeal with the middle class.
Asked what Mr Sunak has to do to win, Mr Davis told Sky News: “I think you’ve got to win the arguments, and the arguments that matter, which is why the 7 per cent matters.
“That’s why that matters. It’s very important. The Tory Party, generally speaking, is a bit older than average. It’s a little bit more middle class, but not so much these days, but a little bit more middle class.
“It will care about things like their offspring having to face these sorts of interest rates in the future, so that matters.
In a rare sign of harmony, both candidates said they would want the other to be involved in their government.
But allies of Ms Truss were reported to have said Mr Sunak had demonstrated “aggressive mansplaining and shouty private school behaviour” during the debate.
Ms Truss described herself as “naturally a thrifty person” when asked about ways to help tackle climate change.
She said she would not give Mr Sunak fashion advice, describing him as a “very well-dressed man”, when asked about criticism made by Culture Secretary Nadine Dorries about his expensive taste in clothing.
Ms Truss was also asked about her suggestion that her comprehensive school background would make her a better prime minister.
Mr Sunak said he was “certainly not going to apologise” for his parents working hard to send him to a private school.
“Winchester is a very, very good school, and I would love people from right across the country to have the opportunity to go to a school like that,” Ms Truss said.
The studio audience in Stoke was made up entirely of people who voted Conservative at the last general election and applauded more often for Mr Sunak.
With postal ballots set to arrive on Tory members’ doorsteps by August 5, Mr Sunak was under pressure to use the BBC debate and another hosted by TalkTV and The Sun on Tuesday to make an early breakthrough.
Meanwhile, Lord Cruddas of Shoreditch, a former Conservative Party treasurer, has said that current Prime Minister Boris Johnson “does not want to resign”.
He said the comments were made to him by Mr Johnson over lunch at Chequers on Friday.
The peer is running a campaign to give Conservative Party members a vote on whether to accept Mr Johnson’s resignation as leader.
Mr Johnson decided to resign after losing the confidence of his MPs.
Downing Street responded by insisting he will leave the post when a new leader is chosen in September.
“The prime minister has resigned as party leader and set out his intention to stand down as PM when the new leader is in place,” a spokeswoman said.