Iron wisely suggests File::Basename. Here's how it goes,

use File::Basename;
my $filename = "../../path/to/test.txt";
my $basename = basename( $filename, '.txt', '.dat');
print $basename, $/;

__END__
test
[download]

If you omit the extension list arguments, or if $filename's extension is not in the list, basename() will return the basename with extension.

F::B also offers the functions dirname(), which extracts just the directory part, and fileparse(), which returns a list of ($basename, $dirname, $extension)