Read the DBD::Sybase docs carefully. Sybase can return multiple result sets from a query, and most of the system stored procedures do in fact do that. You should structure your result retrieval something like this:

do
{
    while(my $row = $sth->fetchrow_array)
    {
        # do something with the row
    }
} while($sth->{syb_more_results});
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Note that you need to check for syb_more_results after your first call to fetch*.

Update
I forgot to add: some of what dbcc prints isn't even a result set, it's a server message (the output of print). You'll need to install an error handler, as outlined in the docs under "Sybase-specific attributes". There's an example there of how to print showplan output that you can use as a model.