It comes as Alex Burghart, who was visiting Edinburgh University to celebrate the second year of the Turing Scheme, which replaced Erasmus, and Scotland Office minister Iain Stewart backed the Prime Minister over Partygate and said Boris Johnson should not resign even if fined by the Metropolitan Police.
Critics such as Scotland's higher education minister Richard Lochhead said at the time the loss of access to the Erasmus scheme was a "huge blow”, while the SNP said the Tories had “destroyed” one of the “greatest international schemes ever created”.
Mr Johnson had said there was “no threat to the Erasmus scheme” during the Brexit negotiations.
The two ministers were visiting Edinburgh on Thursday to mark the launch of the second year of bidding for Turing scheme placements by universities.
Mr Burghart said: “Despite all the difficulties of the pandemic, despite the fact this was probably one of the worst years imaginable in order to launch a global mobility programme, we’ve seen real success, and the potential of the programme is manifest.
Asked whether losing the programme meant the UK Government had broken a Brexit promise, he added: “In many respects it [Turing] goes further than Erasmus and people who have the chance to go on it will actually have a wider range of choices than they would previous have.
"I think that the Government is delivering on its promise to create a global Britain with global opportunities for its citizens.”
Mr Stewart added: “Turing goes a lot further than Erasmus in many areas and I would encourage any young person, it’s about expanding all those international opportunities to learn new experiences, work with like-minded students elsewhere in the world.”
The SNP/Green coalition has committed to establishing its own exchange programme for higher and further education to replace Erasmus. However, no details of the plans have been made public.
Wales also launched its own scheme, Taith, earlier this year.
An SNP spokesperson said young people’s horizons had been “restricted” by “Boris Johnson’s botched Brexit deal” and that independence would see Scotland rejoin Erasmus.
They said: "Their Erasmus replacement scheme has left Scottish students worse off, out of pocket, and in eye-watering levels of debt. They have, quite simply, destroyed one of the greatest international schemes ever created.
"As an independent country, Scotland will have the opportunity to take back its place within Europe as an equal partner, which in turn will offer a pathway back into the Erasmus programme."
Both ministers also threw their weight behind the Prime Minister as he battles demands to resign over Partygate.
The first 20 fines for the illegal gatherings in Downing Street were issued this week, but both Mr Burghart and Mr Stewart said Mr Johnson should stay in post despite the law-breaking activities.
Mr Burghart said: “The Prime Minister is doing a fantastic job in one of the most challenging times that any Prime Minister since 1945 has had to deal with and he has got my full confidence.”
Mr Stewart added: “When you look at all the big calls, he’s got them right.”
Scottish higher education minister, Jamie Hepburn, said: “The UK Government’s Turing Scheme does not match the breadth and scope of Erasmus+ and will see support for our most disadvantaged learners cut, and opportunities for all our students, staff and young people reduced.”
He added the Scottish Government is still developing its replacement for Erasmus.