Group B winners Wales meet Northern Ireland, one of the tournament's four best third-placed teams, in the last 16 in Paris on Saturday evening and find themselves in what is seen as the easiest half of the draw in France.
England, meanwhile, are in the other half of the draw and have a potential quarter-final against hosts France with Germany, Spain and Italy possibly waiting for them in the last four should they get past Iceland.
"It is a moral victory for us," said Bale.
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"Bragging rights to finish top. It is always nice to get one over on the English."
Wales' Gareth Bale celebrates after the match
Image credit: Reuters
Wales lost 2-1 to England in Lens last week, but their emphatic 3-0 victory over Russia and the failure of Roy Hodgson's side to beat Slovakia saw them top the group.
Belgium would be expected to meet the winners of Wales and Northern Ireland in the last eight, while Croatia are potential semi-final opponents in a tournament which now looks extremely open.
But Bale - who became the first player since 2004 to score in all three group games at the Euros - insists he is looking no further than the all-British affair at the Parc des Princes.
"Obviously you come to the tournament for one reason," said Bale, the competition's joint top scorer with three goals.
"To win. Not to play three games and go home. The ultimate goal is we want to try and win the tournament.
"Yes, we would love to win it, but all our focus now is on Northern Ireland.
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Image credit: PA Sport
"Until that whistle goes and we're through we can't think about it a second before.
"We can't think about Belgium. We know our side of the draw.
"Maybe it is more favourable. But it is not favourable if you don't get through."
Northern Ireland drew 1-1 in a Cardiff friendly in March ahead of the Euros.
But Wales were severely depleted by injury that night with Bale and Aaron Ramsey among the absentees.
Wales were set to play Turkey until the Republic of Ireland's late winner against Italy on Wednesday night changed the dynamics of the second round draw.
"At this stage no matter who you get it is not going to be an easy game," said Bale.
"We're happy to be in the round of 16, whether it's against a team that normally wins like Spain or Germany, or a newer one like Northern Ireland.
"We obviously know more about Northern Ireland than Albania or Turkey and we are happy with the draw.
"It will be a British-type game and it will be difficult.
"But if we can match the performance against Russia we have a very good chance."
Bale said there is an "amazing feeling" in camp after the 3-0 victory over Russia.
It was described as one of the best performances in Welsh football history and showed how far Chris Coleman's team has come in recent times.
"We've come through a massive journey," said Bale.
"We've been in some bad places, we were 112th in the world and now we're in the last 16 of the Euros.
"These are the days are to enjoy.
"But the day after the Russia game it was down to business and back on the training pitch."
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