The Scottish government has announced plans to scrap the £2,000 fee paid by students after graduation.
Education Secretary Fiona Hyslop told the Scottish Parliament current and subsequent students would not have to pay the endowment.
She intends bringing forward draft legislation in the autumn, with the aim of the move coming into force by April.
Scottish Labour and the Scottish Tories said the statement from Ms Hyslop lacked detail on key issues.
Ms Hyslop told Holyrood that the graduate endowment had "clearly failed".
She claimed that in the three years it had been in operation, two thirds of those due to pay the fee had not done so directly but just added it to their student loan.
"It is clear to me that the graduate endowment fee is a very complicated and inefficient way of generating money for student support," she said.
"Not only does it impact on graduates as a 'back-end' tuition fee, but the law on this actually states that not one penny of the fee can go towards paying for learning or teaching at university.
"This is the worst of both worlds."
Ms Hyslop said Scottish education should be based on the ability to learn, not the ability to pay.
She told MSPs: "It must therefore be wrong to burden our graduates with debt and deny them with every possible opportunity to contribute to a wealthier and fairer Scotland.
"It is a wrong that they begin their working life encumbered by financial pressures and it is a wrong this new Scottish government intends to put right."
Labour education spokesman Hugh Henry branded the statement "meagre and disingenuous", and criticised the government for stating that work on the key SNP manifesto commitments of bringing in means-tested grants and debt repayment relief was "actively continuing".
"It tinkers at the edge of what the SNP promised to students," he said of the statement.
"(There was) no promise in the statement of even a single extra penny of investment in either higher education or further education.
"Why has the SNP failed to tell this parliament how much they are going to invest to allow higher education and further education to compete not just against the rest of the world, but against English institutions?"
Murdo Fraser, the Tory education spokesman, said that while his party was no fan of the graduate endowment, Ms Hyslop had failed to address the real issue facing Scottish higher education, adding that there was more to government than "grabbing a few cheap headlines".
Fiona Hyslop said the endowment had failed
"She must be aware about the growing concern there is among Scottish universities that they face a competitive disadvantage as against English institutions due to the additional revenue derived by top-up fees," he said.
"The real issue is how to fill the funding gap and on this issue the cabinet secretary has not one word to say."
Lib Dem education spokesman Jeremy Purvis, whose party helped bring in the endowment as part of the last coalition government, added: "We welcome the opportunity of working with the minority government to end the graduate endowment as a move towards reducing student indebtedness."
Scrapping the £2,289 endowment, at a cost of about £15m, has gained just enough parliamentary backing to go through, with support from at least the Scottish Liberal Democrats and Scottish Greens.
Up-front university tuition fees in Scotland were abolished under the previous administration.
Students from outside Scotland but who study there pay £1,700 a year in tuition fees, or £2,700 for those studying medicine.