How can I expand a child div to 100% screen width if the container div is smaller?

You can set the width based on the vw (viewport width). You can use that value too using the calc function, to calculate a left-margin for the div. This way you can position it inside the flow, but still sticking out on the left and right side of the centered fixed-width div.

Support is pretty good. vw is supported by all major browsers, including IE9+. The same goes for calc(). If you need to support IE8 or Opera Mini, you're out of luck with this method.

-edit-

As mentioned in the comments, when the content of the page is higher than the screen, this will result in a horizontal scrollbar. You can suppress that scrollbar using body {overflow-x: hidden;}. It would be nice though to solve it in a different way, but a solution using left and rightlike presented in Width:100% without scrollbars doesn't work in this situation.

div {
  min-height: 40px;
  box-sizing: border-box;
}
#parent {
  width: 400px;
  border: 1px solid black;
  margin: 0 auto;
}

#something {
  border: 2px solid red;
}

#wide-div {
  width: 100vw;
  margin-left: calc(-50vw + 50%);
  border: 2px solid green;
}

<div id="parent">
  <div id="something">Red</div>
  <div id="wide-div">Green</div>
  <div id="something-else">Other content, which is not behind Green as you can see.</div>
</div>

After much research, I found this solution: Creating full width (100% ) container inside fixed width container. I think that it is the best solution because it does not depend on any external factor, only the div that you want to expand.

<div class="container" style="width: 750px; margin: 0 auto;">
   <div class="row-full">
     --- Full width container ---
   </div>   
</div>

.row-full{
     width: 100vw;
     position: relative;
     margin-left: -50vw;
     left: 50%;
}

Typically the responsive element, bootstrap or Foundation, allow you to add a "row" element. You can put the "wide-div" outside an element with "row" and it should expand to take up the full width.

Alternatively, you can use absolute positioning for that element which ignores most inherited settings:

.wide-div {
    position: absolute;
    left: 0;
    right: 0;
}