node 10 and newer

Since Node.js v10.0, there is assert.rejects which does just that.

Older versions of node

async functions never throw – they return promises that might be rejected.

You cannot use assert.throws with them. You need to write your own asynchronous assertion:

async function assertThrowsAsynchronously(test, error) {
    try {
        await test();
    } catch(e) {
        if (!error || e instanceof error)
            return "everything is fine";
    }
    throw new AssertionError("Missing rejection" + (error ? " with "+error.name : ""));
}

and use it like

return assertThrowsAsynchronously(aPromise);

in an asynchronous test case.