I just wanted to expand on the accepted answer with support for Internet Explorer (most modern versions, anyways), and to tidy up the code using jQuery:

$(document).ready(function() {
    saveFile("Example.txt", "data:attachment/text", "Hello, world.");
});

function saveFile (name, type, data) {
    if (data !== null && navigator.msSaveBlob)
        return navigator.msSaveBlob(new Blob([data], { type: type }), name);
    var a = $("<a style='display: none;'/>");
    var url = window.URL.createObjectURL(new Blob([data], {type: type}));
    a.attr("href", url);
    a.attr("download", name);
    $("body").append(a);
    a[0].click();
    window.URL.revokeObjectURL(url);
    a.remove();
}

Here is an example Fiddle. Godspeed.

Same principle as the solutions above. But I had issues with Firefox 52.0 (32 bit) where large files (>40 MBytes) are truncated at random positions. Re-scheduling the call of revokeObjectUrl() fixes this issue.

function saveFile(blob, filename) {
  if (window.navigator.msSaveOrOpenBlob) {
    window.navigator.msSaveOrOpenBlob(blob, filename);
  } else {
    const a = document.createElement('a');
    document.body.appendChild(a);
    const url = window.URL.createObjectURL(blob);
    a.href = url;
    a.download = filename;
    a.click();
    setTimeout(() => {
      window.URL.revokeObjectURL(url);
      document.body.removeChild(a);
    }, 0)
  }
}

jsfiddle example

Late, but since I had the same problem I add my solution:

function newFile(data, fileName) {
    var json = JSON.stringify(data);
    //IE11 support
    if (window.navigator && window.navigator.msSaveOrOpenBlob) {
        let blob = new Blob([json], {type: "application/json"});
        window.navigator.msSaveOrOpenBlob(blob, fileName);
    } else {// other browsers
        let file = new File([json], fileName, {type: "application/json"});
        let exportUrl = URL.createObjectURL(file);
        window.location.assign(exportUrl);
        URL.revokeObjectURL(exportUrl);
    }
}

saveFileOnUserDevice = function(file){ // content: blob, name: string
        if(navigator.msSaveBlob){ // For ie and Edge
            return navigator.msSaveBlob(file.content, file.name);
        }
        else{
            let link = document.createElement('a');
            link.href = window.URL.createObjectURL(file.content);
            link.download = file.name;
            document.body.appendChild(link);
            link.dispatchEvent(new MouseEvent('click', {bubbles: true, cancelable: true, view: window}));
            link.remove();
            window.URL.revokeObjectURL(link.href);
        }
    }

This is my solution. From my point of view, you can not bypass the <a>.

function export2json() {
  const data = {
    a: '111',
    b: '222',
    c: '333'
  };
  const a = document.createElement("a");
  a.href = URL.createObjectURL(
    new Blob([JSON.stringify(data, null, 2)], {
      type: "application/json"
    })
  );
  a.setAttribute("download", "data.json");
  document.body.appendChild(a);
  a.click();
  document.body.removeChild(a);
}

<button onclick="export2json()">Export data to json file</button>

Working example of a download button, to save a cat photo from an url as "cat.jpg":

HTML:

<button onclick="downloadUrl(' 'cat.jpg')">Download</button>

JavaScript:

function downloadUrl(url, filename) {
  let xhr = new XMLHttpRequest();
  xhr.open("GET", url, true);
  xhr.responseType = "blob";
  xhr.onload = function(e) {
    if (this.status == 200) {
      const blob = this.response;
      const a = document.createElement("a");
      document.body.appendChild(a);
      const blobUrl = window.URL.createObjectURL(blob);
      a.href = blobUrl;
      a.download = filename;
      a.click();
      setTimeout(() => {
        window.URL.revokeObjectURL(blobUrl);
        document.body.removeChild(a);
      }, 0);
    }
  };
  xhr.send();
}

window.location.assign did not work for me. it downloads fine but downloads without an extension for a CSV file on Windows platform. The following worked for me.

    var blob = new Blob([csvString], { type: 'text/csv' });
    //window.location.assign(window.URL.createObjectURL(blob));
    var link = window.document.createElement('a');
    link.href = window.URL.createObjectURL(blob);
    // Construct filename dynamically and set to link.download
    link.download = link.href.split('/').pop() + '.' + extension; 
    document.body.appendChild(link);
    link.click();
    document.body.removeChild(link);

this is a good easy solution for it.

function downloadBloob(blob,FileName) {
    var link = document.createElement("a"); // Or maybe get it from the current document
    link.href = blob;
    link.download = FileName;
    link.click();
}

If you want to download a pdf and using window.location is not mandatory, you can use jsPdf like this following :

// Create document
const doc = new jsPDF('l', 'px', 'a4');

// [...] Add here the jsPdf doc filling

// Launch the document downloading
doc.output('save', 'filename.pdf');