The former Chancellor said he would “fight for every vote” despite polls of Tory members giving Ms Truss a significant lead.
Speaking at the hustings event hosted by LBC in Leeds on Thursday, Mr Sunak said: “I know the polls say I’m behind in this race. I know there are people who say there should be a coronation, not a contest.
“But I heard that once before seven years ago, when I arrived in Richmond, North Yorkshire, and ultimately the members there gave me the greatest honour of my life when they selected me to be their candidate to be their Member of Parliament.
“I’m asking for all of your support. And I promise you I am gonna fight for every single vote.”
He pledged to “grip” inflation, tackle the NHS backlog and “reunite” the country - but said he would not embark on a “spending spree borrowing tens and tens of billions of pounds of unfunded promises and put them on the country’s credit card”.
“That’s not right. That’s not responsible, and it’s certainly not Conservative. But of course, once we grip inflation and ensure that mortgage rates don’t rise and cripple people, I’m going to cut taxes,” he added.
Responding to an attack on the price of his suit and shoes by culture secretary Nadine Dorries on Wednesday, Mr Sunak said: “It’s not about what shoes or suit I’m wearing, it’s about what I’m going to do for the country”. His response received a loud round of applause from the audience.
Meanwhile, Ms Truss vowed to “get Northern Powerhouse Rail built” and criticised the standard of transport in Leeds and the north of England as she made her pitch to be Prime Minister.
The Foreign Secretary also made reference to her relocation to Norfolk, the site of her constituency, and the fact she has become a supporter of Norwich City FC.
She said “I do want us to channel the spirit of Don Revie” – a former Leeds United and England manager – because “we need to win”.
“And, my friends, we can win against Keir Starmer, who is a patronising, plastic patriot,” she added.
Referencing her upbringing in the city, she said it was “fantastic” to be in her “old stomping ground”, adding that she hoped none of her former teachers were in the audience.
Ms Truss said she got “grit, determination and straight-talking” from Yorkshire, and that is what is needed in Downing Street as “we face a huge global economic crisis”.
She made further references to the area, saying she would keep corporation tax low to encourage investment in towns and cities across Yorkshire, as well as the country as a whole.
During questions from LBC’s Nick Ferrari, Mr Sunak also flatly denied that he would give Boris Johnson a job in the Cabinet.
Asked to comment on reports that some 14,000 members want Mr Johnson’s name on the leadership ballot, he responded: “I’d say to them that I think close to 60 people resigned in Parliament and it’s incumbent on the Prime Minister to have the confidence of the parliamentary party, and that wasn’t there at the end.”
Later on, Mr Sunak was accused of “stabbing” the Prime Minister in the back by a Tory member.
He replied: “I'm proud of this Government and we achieved many great things. But for me personally it got to a point where I couldn't stay. I had significant difference of opinion with him on how to handle the economic challenges that were ahead of us.”
Mr Sunak also backed the return of grammar schools, saying he believed in “educational excellence” and that he would “reform the system to get a better outcome”.
Ms Truss said she had “always been a fan of Boris Johnson” to loud applause from the audience.
“I think he did a fantastic job as prime minister. He delivered Brexit and he delivered the Covid vaccine. I was proud to serve as a loyal member of his Cabinet.”
However, she admitted that she had made a “mistake” by voting with the Government over the Owen Paterson affair, adding: “No, I wouldn’t do it again if I had my time again.”