Mr Williamson said he had “every confidence” the Government will be able to run a whole set of exams in the summer of 2021.
Instead, students were given grades based on teacher assessments moderated by an algorithm.
But the algorithm was quickly discredited, forcing the Government into a chaotic U-turn with marks being recalculated based solely on teacher assessments.
Mr Williamson was asked on Sky News to give a cast-iron guarantee that exams would not be cancelled again.
He said even during the national lockdown there had been exams running and added: “Tens of thousands of students have been taking those GCSE and A-level exams all the way through that national lockdown, and that's been done safely and securely and successfully.
“I have every confidence if we've been able to run a whole set of exams for GCSEs and A-levels during a national lockdown, we have every ability to run those same set of exams in the summer of next year.”
Mr Williamson’s pledge came as he revealed GCSE and A-level students will be awarded more generous grades to compensate for the pandemic.
Mr Williamson said pupils in England will receive advance notice of some topics ahead of tests – as well as exam aids when sitting papers – to ensure this cohort of students is not disadvantaged.
Additional exams will also be run to give students a second chance to sit a paper if the main exams or assessments are missed due to illness or self-isolation, the Department for Education (DfE) said.
In October, the Government announced that the 2021 exams would still go ahead in England, but that the majority of them would be delayed by three weeks to give pupils more time to catch up on learning.
Students will be given aids, such as formula sheets, in some exams to boost their confidence and reduce the amount of information they need to memorise, as part of the measures.
But it is understood that grading changes simply based on the region you live in have been ruled out.
Under new contingency measures, students who miss one or more exams due to self-isolation or sickness, but who have still completed a proportion of their qualification, will still receive a grade.
If a student misses all their assessments in a subject, they will have the opportunity to sit a contingency paper held shortly after the main exam series.
These tests are expected to run in the first few weeks of July.
If a pupil has a legitimate reason to miss all their papers, then a validated teacher-informed assessment can be used but only once all chances to sit an exam have passed.
Students who are clinically extremely vulnerable will also be given the option to sit an exam at home if they cannot be in school due to restrictions.
It comes after DfE figures revealed that 22 per cent of secondary school pupils were absent from school last week for the second week running.
Mr Williamson said: “Exams are the best way of giving young people the opportunity to show what they can do, which is why it’s so important they take place next summer.
“But this isn’t business as usual. I know students are facing unprecedented disruption to their learning. That’s why exams will be different next year, taking exceptional steps to ensure they are as fair as possible.
“I am determined to support students, parents and teachers in these unprecedented times and hope measures like more generous grading and advance notice of some topic areas will give young people the clarity and confidence they need to achieve every success.”
The DfE has also announced that full, graded Ofsted inspections will not resume until the summer term and exam results will not be included in school performance tables this year.
Sats exams in Year 6 will still go ahead – except for the grammar, punctuation and spelling test – but tests in Year 2 will be suspended for a year.
Ofqual’s interim chief regulator Dame Glenys Stacey said: “Summer 2020 results were the first pandemic results. They were unique when compared to previous years, with higher grades overall.
“We have decided to carry forward the overall level of generosity from 2020 through to summer 2021, in recognition of the baleful and continuing impact of the pandemic.
“This is an unprecedented step. Having consulted widely, we think it the right thing to do.”
It is understood the level of generosity will be evened out across subjects to prevent significant differences in the number of students awarded top grades depending on the subject.
But James Turner, chief executive of the Sutton Trust, said the measure would need “careful management” to ensure it does not widen existing attainment gaps “as students at more affluent schools may have better access to the resources to prepare these topics in detail and at short notice”.