Rishi Sunak has pledged to "work closely" with Nicola Sturgeon "to respond to the shared challenges we face".
The new Prime Minister spoke with the SNP leader on a phone call on Tuesday evening at the end of his first official day in 10 Downing Street.
The First Minister described the call as "constructive" and said she hoped there would be "further engagement soon".
Sunak had earlier become the latest Tory MP to take the top job in Westminster after formally meeting with the King at Buckingham Palace.
After completing a Cabinet reshuffle he began a round of calls with international and devolved leaders around 9pm.
In a tweet, Sunak said he had spoken to Sturgeon and Welsh First Minister Mark Drakeford.
He said: "I emphasised our duty to work closely together to respond to the shared challenges we face, so that collectively we can deliver for the people of the United Kingdom."
Sturgeon tweeted a short time after: "I expressed hope that we will build a UK/Scot government relationship based on mutual respect - including for mandates - and my fear that further austerity will do real damage to people and public services.
"I look forward to further engagement soon."
A Scottish Government spokesperson said: "The First Minister and the Prime Minister had a brief, introductory call this evening. The First Minister congratulated the Prime Minister on his appointment and wished him well.
"She expressed her hope that political differences notwithstanding, they would build a constructive working relationship. She made clear that the Scottish Government would do everything possible to establish such a constructive relationship but stressed that this must be built on mutual respect.
"The First Minister also expressed the strong view that the UK Government should address the pressure and pain being felt by people and businesses as a result of inflation and other economic pressures, and should not exacerbate that with a further wave of austerity.
"She welcomed the Prime Minister’s assurance of engagement with the devolved Governments ahead of the Chancellor’s forthcoming budget statement.
"Finally, the First Minister made clear her intention to honour the mandate the Scottish Government received from the people of Scotland at the last election."
Relations between Holyrood and Westminster have plummeted in recent months after former PM Liz Truss declared the best way to deal with the First Minister was to "ignore her".
Truss failed to hold a single formal meeting with Sturgeon during her disastrous 50-day period in 10 Downing Street.
Sturgeon previously told the BBC she had not had a conversation with Truss since she was appointed on September 6, other than "an exchange or two" at events following the death of the Queen.
"It’s quite absurd in many ways. When I became First Minister, David Cameron was prime minister and I think we spoke on the phone the first night I became First Minister,” she said.
"I spoke on the phone to Theresa May within a day or two of her becoming prime minister, same with Boris Johnson actually.
"I have deep political differences with all of these politicians, but we have a duty to work together constructively."
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