Is there any 'Auto-complete polygon' tool in QGIS?

Yes, you can set snapping tolerance (e.g. to vertex and segment) you can then over digitize. A new polygon will only be created to the edge of an existing polygon. This setting can be enabled through the Settings Menu> Project Properties> General tab.

Update for versions 3.x

From the Project > Snapping Options dialogue:

1. Enable snapping (button with magnet icon)
2. Enable 'topological editing' in the next button
3. Switch 'all layers' to 'advanced configuration' in the next button
4. For the layer you want auto-complete polygon to work with, ensure the layer is checked on
5. Enable the 'avoid overlap' option for that same layer

Now digitize a new polygon ensuring you are either starting and stopping the sketch inside the original polygon, or at least crossing the original polygon where you want to auto-complete your new polygon along: