Do you have any pictures? How large are the leaves? What zone are you in?

There are so many types of vines, it is hard to identify it based on it being thin.

Generally though, what you'd want to do is not pick it off the flower, but find out where it is growing from and dig it up, you possibly may have to dig it up in multiple locations. You can also try spraying it with a herbacide like Round Up if there is a patch of it that isn't touching or near your flowers (you do not want to accidentially spray your flowers).