If it's object-oriented, you could use a singleton pattern. (There's a non OO approach that looks very similar.)

my $single;
sub new {
    return $single if defined $single;
    # set up args
    $self->{dbh} = DBI->connect(@args);
    $single = bless($self, $class);
    return $single;
}

# procedural 
{
    my $dbh;
    sub get_handle {
        return $dbh if defined $dbh;

        # handle arguments
        $dbh = DBI->connect(@args);
    }

}
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