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:
- Enable snapping (button with magnet icon)
- Enable 'topological editing' in the next button
- Switch 'all layers' to 'advanced configuration' in the next button
- For the layer you want auto-complete polygon to work with, ensure the layer is checked on
- 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: