The decision means that no exams will take place in Scotland next Summer after the decision earlier this year to axe National 5s.
Teaching representatives welcomed the decision, but one Scots parents group warned it will have a “terrible impact” on pupils’ futures.
The Education Secretary announced the cancellation of Highers at Holyrood after growing concerns that too many pupils have unfairly missed out schooling as a result of being ordered to self isolate.
Youngsters will now receive grades based on teacher assessments of classroom work throughout the year.
"There will be no Higher or Advanced Higher exams in 2021,” Mr Swinney told MSPs in a statement.
“Instead we will adopt the new model that has been developed and base awards on teacher judgement of evidence of learner attainment.
“This is safe. It is fair.
“And it better recognises the reality of the disruption so many pupils have already had to their learning.”
Mr Swinney had previously indicated that he could have waited until the end of next February before making a decision on the issue, but has faced growing pressure from opposition parties to “end the uncertainty” for pupils and teachers going into the festive period.
He added: “I will not stake the future of our Higher pupils – whether they get a place at college, university, training or work – on a lottery of whether their school was hit by Covid.
“Exams cannot account for differential loss of learning and could lead to unfair results for our poorest pupils.
“This could lead to pupils’ futures being blighted through no fault of their own.
“That is simply not fair.”
It will mark the second year in a row in which Scotland’s exams diet has been cancelled after the Covid pandemic saw it axed this year.
A special bonus payment will also be made to teaching staff this year in acknowledgement of the “additional workload of assessment of national qualifications in this unique academic year” with the exams being axed.
" I intend to make an exceptional one off payment to teachers and lecturers who are critical to assessing and marking N5, Higher and Advanced Higher
courses this year,” Mr Swinney added.
"We will progress this urgently with partners and employers, including discussing when and how the payment will be delivered.”
Secondary schools are now being urged to prioritise all remaining in-service day time to work on the alternative model of certification for the national qualifications next year.
The decision follows a similar move by the Welsh Government to cancel their exam diet for 2021 in November while the government in England confirmed their exams would go ahead in May.
Concerns have also been raised about the disproportionate impact on poorer and older pupils caused by Covid-19 with those groups more likely to miss school due the impact of the virus and requirements to self-isolate.
In 2020, the exam results were plunged into chaos when it emerged 125,000 teacher estimates were moderated down based on a statistical model used by the SQA, leading to an embarrassing u-turn for the Scottish Government and all teacher estimates being awarded to pupils.
But Mr Swinney told MSPs that the new system which has been devised for grade estimates next year will not include the controversial "algorithm" which resulted in many pupils being marked down this year.#
"Let me make clear that no algorithm will be used," he said.
"It is a model that will be based on learner evidence, subject to quality assurance at local and national level, to deliver a credible and fair set of results.
"It is a model that has achieved a broad level of support across Scotland’s education professionals."
The methodology behind the new system has been set out in a report by the National Qualifications 2021 Group, he added.
Larry Flanagan, general secretary of the EIS teaching said cancellation was the right move.
“The EIS has every confidence in the ability of teachers to make professional judgements based on pupil evidence and in the circumstances believes that cancelling the exam diet in favour of an alternative model is the correct decision, one which could have been made earlier,” he said.
"We have raised repeatedly, however, the additional workload burden which this will generate and made clear that teachers should not be treated as unpaid SQA markers. The Deputy First Minister’s commitment, therefore, to make additional payments to teachers is welcome.”
But the cancellation came under fire from parent group UsFor Them Scotland.
Organiser Jo Bisset said: “This is an inexcusably bad decision which will have a terrible impact on children’s future.
“John Swinney made it abundantly clear that exams would go ahead if public health guidance allowed.
“Instead, it seems like he’s now taking his orders from vested interests and politicians who have, at best, base level ambitions for Scotland’s pupils.
Tory education spokesman Jamie Greene accused the Mr Swinney of a lack of "leadership" over the decision.
“John Swinney delayed this decision to the last possible moment, shamefully keeping pupils, teachers and parents in the dark for months,” he said.
“At every turn during this pandemic, he has failed to step up and show the leadership required.”
But the cancellation of the Highers was welcomed by Greens education spokesman Ross Greer who described it as "overdue."
He added: “It provides the clarity that teachers, parents and, most importantly, pupils had demanded.
“What’s essential now is that the Education Secretary stops the SQA repeating its approach to National 5 assessments with the Higher and Advanced Highers.
“Despite Mr Swinney’s categorical assurances to me earlier this year, the SQA has created a system which has massively added to teachers’ workloads, essentially expecting them to take on the huge additional work of an SQA marker.”