By my reading of the IO::Compress::Gzip documentation, you could do something like this:

open my $fh, '<', $originalFile
    or die "Can't read '$originalFile': $!\n";
IO::Compress::Gzip::gzip $fh => $gzippedPath;
close $fh or die "Close failed on '$originalFile: $!\n";
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Update: Upon further reading, it seems you can use optional parameters to override the name or write a gzip file without that data.

# No name, no time (nameless and timeless?)
IO::Compress::Gzip::gzip $originalFile => $gzippedPath,
    Minimal => 1;
# A name of my choosing.
IO::Compress::Gzip::gzip $originalFile => $gzippedPath,
    Name => 'Nyuck' x 3;
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