Just a test which is working fine.

#parent{
    width: 100%;
    height: 100%;
    overflow: hidden;
}

#child{
    width: 100%;
    height: 100%;
    overflow-y: scroll;
    padding-right: 17px; /* Increase/decrease this value for cross-browser compatibility */
    box-sizing: content-box; /* So the width will be 100% + 17px */
}

Working Fiddle

JavaScript:

Since the scrollbar width differs in different browsers, it is better to handle it with JavaScript. If you do Element.offsetWidth - Element.clientWidth, the exact scrollbar width will show up.

JavaScript Working Fiddle

Or

Using Position: absolute,

#parent{
    width: 100%;
    height: 100%;
    overflow: hidden;
    position: relative;
}

#child{
    position: absolute;
    top: 0;
    bottom: 0;
    left: 0;
    right: -17px; /* Increase/Decrease this value for cross-browser compatibility */
    overflow-y: scroll;
}

Working Fiddle

JavaScript Working Fiddle

Information:

Based on this answer, I created a simple scroll plugin.