Arsenal midfielder Ramsey scored the opening goal as Wales recorded a 3-0 win over Russia on Monday, a result that clinched top spot in Group B.
Neil Taylor and Gareth Bale also found the net as Chris Coleman's side made sure they extended their stay on French soil.
"We wanted to come to this competition and really make a statement, put Welsh football on the map and show what we are capable of doing. I think we've done that," he told ITV.
"Maybe against England we let the occasion get to us a bit, in terms of when we had the ball. That was something we really wanted to improve on [against Russia].
"We've shown what a good defensive team we are, but now we have shown what a good team we are in possession. We created loads of opportunities and we can really be proud of what we have achieved so far."
For only the second time in their history, Wales are through to the knockout stages of a major international tournament.
They do not yet know the identity of last-16 opponents, although they are certain to face a third-placed team from either Group A, C or D in Paris on Saturday.
"That was our main target - to get out of the group," Ramsey added. "We've achieved that and finished top.
"It’s a great day for Welsh football. I've been waiting for a while for this moment, to show what we are capable of on a big stage.
"Who knows what can happen now?"