I thought I'd add another solution of doing this, using behaviors from the Expression SDK and combining it with the solution from @Thomas. Using that, we can define a "CloseBehavior" that handles the code behind of starting a storyboard and closing the window when it's done.
using System.ComponentModel;
using System.Windows;
using System.Windows.Interactivity;
using System.Windows.Media.Animation;
namespace Presentation.Behaviours {
    public class CloseBehavior : Behavior<Window> {
        public static readonly DependencyProperty StoryboardProperty =
            DependencyProperty.Register("Storyboard", typeof(Storyboard), typeof(CloseBehavior), new PropertyMetadata(default(Storyboard)));
        public Storyboard Storyboard {
            get { return (Storyboard)GetValue(StoryboardProperty); }
            set { SetValue(StoryboardProperty, value); }
        }
        protected override void OnAttached() {
            base.OnAttached();
            AssociatedObject.Closing += onWindowClosing;
        }
        private void onWindowClosing(object sender, CancelEventArgs e) {
            if (Storyboard == null) {
                return;
            }
            e.Cancel = true;
            AssociatedObject.Closing -= onWindowClosing;
            Storyboard.Completed += (o, a) => AssociatedObject.Close();
            Storyboard.Begin(AssociatedObject);
        }
    }
}
The behavior defines a storyboard as a dependency property, so we can set it in xaml and when the AssociatedObject (the window where we define the behavior) is closing, this storyboard is started using Storyboard.Begin(). Now, in xaml we simply add the behavior to the window using the following xaml
<Window x:Class="Presentation.MainWindow"
        xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
        xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
        xmlns:behave="clr-namespace:Presentation.Behaviours"
        xmlns:i="clr-namespace:System.Windows.Interactivity;assembly=System.Windows.Interactivity"
        x:Name="window">
    <Window.Resources>
        <Storyboard x:Key="ExitAnimation">
            <DoubleAnimation Storyboard.Target="{Binding ElementName='window'}"
                             Storyboard.TargetProperty="(Window.Opacity)"
                             Duration="0:0:1" From="1" To="0"/>
        </Storyboard>
    </Window.Resources>
    <i:Interaction.Behaviors>
        <behave:CloseBehavior Storyboard="{StaticResource ExitAnimation}"/>
    </i:Interaction.Behaviors>
    <Grid>
    </Grid>
</Window>
Note the xml namespace i from the System.Windows.Interactivity dll, and also that the window is referenced, so it has to have a x:Name assigned. Now we simply add the behavior to every window on which we wish to execute a storyboard before closing the application, instead of copying the logic to every code-behind in each window.