The prime minister has called a summit with the leaders of the devolved administrations in Scotland, Wales and Northern Ireland, saying the UK is "best served when we work together".
The SNP leader said Mr Johnson would be "picking a fight with the democratic wishes of the Scottish people" if he tries to block another vote.
Watch: Sturgeon warns Westminster not to block independence push
Although Ms Sturgeon fell one seat short of an overall majority, pro-independence parties will hold a majority of seats at Holyrood.
In the wake of his victory, Mr Drakeford called on the PM to "reset relationships" with the nations of the UK.
He urged Mr Johnson to carry out a "serious examination of the way in which we can create the machinery that will allow us to work together in the future".
"Not an approach that thinks flying more Union Jacks at the tops of buildings, but proper, respectful relationships that recognise that sovereignty is now dispersed across four parliaments in which we choose to pool it for common purposes," he added.
The PM has written to Ms Sturgeon and Mr Drakeford to invite them to talks to "discuss our shared challenges" in a bid to work together in what he called "Team UK".
Mr Johnson will be sending a similar letter to Northern Ireland's first and deputy first ministers.
"I believe passionately that the interests of people across the UK... are best served when we work together," the PM said.
"I would like to invite you to join me, UK government colleagues and others at a summit meeting to discuss our shared challenges and how we can work together in the coming months and years to overcome them.
"We will all have our own perspectives and ideas - and we will not always agree - but I am confident that by learning from each other we will be able to build back better, in the interests of the people we serve."
Mr Johnson said the UK needed to "show the same spirit of unity and cooperation that marked our fight against the pandemic" in ensuring the Union recovers from COVID-19.
He said it would be a "difficult journey" and added: "The broad shoulders of the UK have supported jobs and businesses the length and breadth of the country, but we know that economic recovery will be a serious shared responsibility."
Meanwhile, the PM has promised to "look forward" beyond the pandemic and continue to focus on his "levelling up agenda" after his successful further dismantling of Labour's so-called "red wall".
"Voters have put their trust in Conservative representatives, councillors and mayors and we must deliver for them," he said.
As part of his legislative proposals, the PM will bring back the Police, Crime, Sentencing and Courts Bill, designed to boost police powers, and the Environment Bill, which will set legally binding environmental targets in the run-up to the COP26 summit in Glasgow later this year, in the Queen's Speech.
However, there has still been no confirmation on whether Mr Johnson's key social care reforms, as promised when he was elected in 2019, will be detailed in Tuesday's speech.