Holyrood education secretary Richard Lochhead and his Welsh counterpart Kirsty Williams signed the letter to Gavin Williamson, who was appointed UK education minister following Boris Johnson's victory in the Conservative leadership contest last month.
It calls on the UK Government to continue to participate in the scheme in the event of leaving the EU without a deal in place.
Mr Lochhead and Ms Williams state in their letter that leaving with no deal - and without an alternative Third Country agreement or other arrangement being reached by the UK - universities, colleges, and schools across the UK would be ineligible to submit applications to participate in the final year of the current Erasmus+ programme in 2020.
Between 2014 and 2018, it is estimated more than 15,000 students and staff from Scotland took part in the EU-led scheme, which allows funded temporary study overseas as part of their Scottish courses.
Mr Lochhead said: "Thousands of Scottish students benefit from Erasmus+ yearly, proportionally more than from any other country in the UK.
"The Scottish and Welsh Governments are clear that we must remain a full participant in Erasmus+.
"I am also alarmed to hear the UK Department for Education could be considering an Erasmus+ replacement programme for England only - with potentially no consequential funding for devolved administrations to put in place their own arrangements.
"That's why we have written to the UK Government calling for urgent action and assurances that Scottish students won't miss out."
He added: "It is the Scottish Government's preference to remain in the EU but in the event of a damaging no-deal Brexit, students could now see the door to this fantastic cultural and educational exchange slammed shut.
"It is unacceptable that with less than 12 weeks left until the UK Government plans to take the United Kingdom out of the EU without an agreement in place there is still no plan for alternative arrangements."
A spokeswoman for the Department for Education said: "This Government is working to negotiate a new deal but in the event the UK leaves the EU without an agreement in place we have already guaranteed cover for the payments for successful UK applicants for Erasmus+ and ESC bids.
"Successful bids are those that are approved directly by the European Commission or by the UK National Agency and ratified by the European Commission.
"The UK Government has also repeatedly made clear that it values international exchange and collaboration in education, which is why we are exploring participation in the successor scheme and preparing for a range of potential outcomes."