The Development Team has completed all the stories that were planned for a particular Sprint.
In this case, the team usually decides to speak with the Product Owner to select other stories from the Product Backlog (possibly related to the current Sprint Goal) that can fit within the remaining Sprint time.
It is important to underline that, if the Development Team is not able to complete the selected additional stories, the Sprint will not be compromised.
I want to add that, if the team finishes early and works on additional tasks in several sprints, this is a sign that the team is, for some reasons, under committing (when this happens, it is usually because the stories are overestimated).
The burn-down chart should always reflect the work of the team during the sprint.
When a team is over delivering I prefer to keep the burn-down chart as it is, showing that the team has arrived to 0 (that is to say, completed all the stories in the Sprint Backlog) before the ending of the Sprint.
I would avoid to modify the burn-down chart as you proposed. In my opinion, management can misinterpret such a chart ("Was the team committed to complete 100 or 105 story points?"). To keep track of the additional stories I usually add another chart to the Scrum board: a burn-up chart (the curve goes up, instead of down) that I use to track how much work the team has completed in the Sprint.