The Scottish First Minister declared she had had a "cordial and constructive" first meeting with Mr Sunak, who only took office last month.
He tweeted later that it was "great" to meet the First Minister, adding that: "Teamwork, absolute focus and collective effort will be required to deal with the shared challenges faced by people across the UK."
While Ms Sturgeon said she had "deep political disagreements" with the Conservative, she also stressed the two leaders had an "obligation to work together in the interests of the people we serve".
The talks between the two leaders came as the cost of living crisis continues across the UK, with Ms Sturgeon using the discussions to challenge Westminster to provide "support for the most vulnerable".
With nurses in Scotland having voted in favour of strike action, she demanded more cash from the UK Government for the NHS, saying ministers in Scotland were now "at the limit of what we can do with what is effectively a fixed budget".
She also raised her plans to hold a second Scottish independence referendum, telling Mr Sunak she intends to "honour that mandate" she has for such a vote "with or without the UK Government’s agreement".
With the UK Supreme Court currently considering if the Scottish Government can hold an advisory referendum without the consent from Westminster, Ms Sturgeon was clear she would act "constitutionally and legally".
But she insisted "the right thing to do would be for the two governments to sit down and agree a process" that would allow the Scottish people to decide the matter.
"I made it clear to him if he’s willing to do that, then I will have that discussion with him," Ms Sturgeon said.
She was speaking from Blackpool, where she is due to attend a meeting of the British-Irish Council on Friday, with the Scottish First Minister claiming in recent years there has been a "real lack of respect" from the UK Government for the devolved administrations.
She stated: "If we are to reset and do things differently, it will require change in substance, not just change in words."
However Liz Truss, who was only in charge at Downing Street for a matter of weeks, left office without having spoken to the Scottish First Minister.
With Mr Sunak having taken over from Ms Truss, the First Minister spoke of her "hope we can have a good relationship".
The SNP leader added: "I think we’ve both got a duty to try to achieve that. But it has to be substance, not just rhetoric.
"Over the period, particularly since the Brexit referendum, the UK Government has ridden roughshod over the Scottish Parliament, Welsh Parliament… and there’s been a real lack of respect at times bordering on contempt, and so if the relationship is to improve we have to see a significant change in attitude from the UK Government.
"I hope that’s what we’ll see, certainly he’s saying all the right things in that regard but the proof of the pudding will be in the eating."
Here she said Mr Sunak had to show "not just a willingness to listen and then ignore it all when he leaves the room, but actually a willingness to act as well".
A Scottish Government spokesperson said afterwards Ms Sturgeon and Mr Sunak had a "constructive and cordial discussion during their bilateral meeting in Blackpool".
The spokesperson said: "The two leaders discussed a range of topics, including the need to work together on the cost of living crisis.
"Ahead of the Chancellor’s Autumn Statement the First Minister emphasised the need for benefits to rise in line with inflation to support the most vulnerable and for substantial additional investment to be made by the Chancellor in the health service.
"Given the extreme pressure being faced by public services like the NHS, the First Minister said additional funding for devolved governments was urgently needed to support people, services, and fair public sector pay uplifts, following the double-digit inflation that has eroded the Scottish Government’s budget since its allocation last year when inflation was at 3%.
"The First Minister made clear she intends to honour the manifesto commitment and the electoral mandate for an independence referendum and that the Scottish Government remains ready and willing to negotiate a process to enable that to happen."
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