Is there any 'Auto-complete polygon' tool in QGIS, similar to the one existing in ArcGIS?
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.
From the Settings > Snapping Options dialogue:
Under 'Snapping Mode' choose 'Advanced'
1) Check the box next to the layer you want to 'auto-complete' to (to enable snapping)
2) Check the box under 'Avoid Intersections'
3) Check the box at the bottom 'Enable Topological Editing'
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.