The minimum number of wins needed for a team in the first stage to guarantee is advancement to the next stage is:

From the given information we can complete the following table it is clear that if a team wins 5 games, then also there is no guarantee of its advancement to the next stage, since only 4 teams can go to the next stage.
Note: In the table W → Wins, L → Loose, × → No match (as example there can not be match between 1& and 1, 2 and 2 and so on)
The above is one of such combination. Since after winning 5 matches too, there is no guarantee to advancement, so the answer must be 6, because no two teams can get 7 points each.