She has vowed that next year’s events will be “a celebration of culture without borders” which would send out a powerful message about the outward-looking nature of modern-day Scotland.
Ms Hyslop, who is also responsible for tourism and external affairs, said that the festivals would play a key role in underlining Scotland’s commitment to the rest of Europe in 2017.
The Edinburgh International Festival (EIF) was instigated in 1947 to provide “a platform for the human spirit” after the Second World War.
Both the Fringe – now the biggest cultural celebration in the world – and the film festival were launched in the same year.
Earlier this month veteran impresario Richard Demarco said he believed Brexit represented the biggest ever threat to the Edinburgh festivals.
EIF director Fergus Linehan has revealed that the referendum result had already triggered a rethink of his plans for 2017 but said the event would reflect its 1947 origins, when it was “part of the whole reconstruction of Europe”.
Ms Hyslop said a strong commitment to “internationalism” had already been regularly reinforced by audiences, artists and overseas governments throughout this month’s festivals.
And she insisted the Scottish Government would be doing everything it could to encourage overseas performers and companies to bring work to Edinburgh, as well as help home-grown artists take their work across Europe.
Ms Hyslop said: “There has already been a reinforcement and a recommitment to our internationalism during the festivals.
“I think we’ve seen that from the responses from audiences, artists and other governments giving a firm commitment to support the festivals going forward.
“Culture and creativity clearly relies on international connections. The creative spirit knows no boundaries. This is a celebration of togetherness, rather than the otherness of the politics of the world.
“We’re looking forward to the 70th anniversary, which will be a key contribution to the European message from Scotland and also from the world to Scotland. I’m more enthusiastic than ever before about the internationalisation and celebration of culture without borders as we go into next year.
“There’s absolutely no danger of next year’s festival being diminished by Brexit. I think the result of the vote will reinforce our efforts to connect international and culturally rather than undermine them.
“Scottish culture already is outward-looking and international. There’s been an increasing internationalisation of Scottish culture in the last few years. I expect that momentum to continue.
“I think that has been reinforced by the desire to express internationalism as a result of the EU referendum. But there has to be an inward focus as well as an outward one.”