The Institute for Public Policy Research (IPPR) made several suggestions as to how ministers could take advantage of the Covid-19 crisis, which has already seen a drop in oil extraction.
But it said the UK Government must take responsibility for the "overwhelming majority of the costs of transition" - as Westminster had "received the overwhelming majority of oil and gas revenues".
The think-tank wants the two administrations to work together to "set clear five-yearly targets" to cut the amount of oil and gas extracted, consumed in and exported from the UK to help achieve net-zero emissions by the Scottish and UK government targets of 2045 and 2050 respectively.
It also suggested the law should be changed to cap extractions to prevent the UK from exporting surplus fuels that generate carbon emissions abroad.
By acting now, the IPPR said ministers in Scotland and London could "show global leadership and demonstrate the economic, environmental and reputational benefits of being a major historical oil and gas producer to announce a timetable to wind down fossil fuel production to align with net zero".
The Net Zero North Sea report stressed: "We must reshape the oil and gas industry, to help us achieve net zero emissions and restore nature in Scotland and across the UK.
"Doing so will also require a long-term plan for reducing oil and gas extraction from UK waters over the coming years."
It added: "This is not about managing decline, but a managed transition - from the reliance on fossil fuels, to the creation and expansion of the industries and jobs of the future."
The think-tank stressed "the right investment and support" must be provided for the communities, businesses and people most affected.
With much of the UK oil and gas industry centred on Aberdeen, it called for both the UK and Scottish governments to increase funding to the Aberdeen City Deal, to help create a Low Carbon Wealth Fund for the city and the wider area.
And the report suggested fines should be imposed by governments on those that "continue to delay the decommissioning of their wells", with cash raised from these also being invested in the Low Carbon Wealth Fund.
The report added: "While the Scottish Government has begun to introduce many initiatives that start to grapple with this challenge, much more work is needed in Scotland and from the UK Government who have yet to start fully incorporating the need for a just transition into policymaking."
It was published ahead of the COP26 global climate change conference, which is taking place in Glasgow in November 2021.
Luke Murphy, IPPR associate director, said: "It's time for the UK to move on from oil and gas to a net zero North Sea and a greener and brighter future.
"As host of Cop 26, the UK has the opportunity to lead by the 'power of our example' by committing to keep fossil fuels in the ground and offering a blueprint for affected workers and communities to make the most of the huge opportunities offered by the zero-carbon economy."