I assume your Save button is an ApplicationBarButton (not a normal button). For normal buttons it will just work because they take focus and hence the data-binding will kick in.
For ApplicationBarButtons on the phone it’s a little different because they don’t take focus away from the client app. To ensure the data-binding kicks in when your Save button is clicked, you can add the following code in your handler:
object focusObj = FocusManager.GetFocusedElement();
if (focusObj != null && focusObj is TextBox)
{
var binding = (focusObj as TextBox).GetBindingExpression(TextBox.TextProperty);
binding.UpdateSource();
}