In pure javaScript, getting an element by className is ugly.  See How to Get Element By Class in JavaScript? for more information.
Basically, you'll want this function:
function getElementsByClass(tagType, className) {
    var elems = document.getElementsByTagName(tagType);
    var returns = [];
    for (var i in elems) {
        if ((' ' + elems[i].className + ' ').indexOf(' ' + className + ' ') > -1) {
            returns.push(elems[i]);
        }
    }
    return returns;
}
Once you have that, the rest is not too bad:
var dynamic = document.createElement("div");
dynamic.innerHTML = "this is my contact number.";
var elements = getElementsByClass("div", "fawas");
if (elements.length > 0) {
    // just change the first, as you did in your post
    elements[0].parentNode.insertBefore(dynamic, elements[0].nextSibling);
}
I dynamically create your new div, rather than just a text string.
Then, I get the parent of the element you want to insert after, use the insertBefore function on whatever is after the element of your choice.
Here is a working fiddle.
As others have shown you, this can be a lot cleaner using jQuery.  I don't know enough node.js to know if it has functions that will be more convenient, so I gave a pure JS solution.