DBI::quote is probably overkill. Use placeholders.

my $sql = 'INSERT INTO person VALUES (?, ?, ?)';
my $dbh = DBI->connect(YOUR CONNECTION DETAILS)
  or die DBI::errstr;
my $sth = $dbh->prepare($sql);

my $csv = 'test.csv';
open my $data, '<', $csv or die "Cannot Open File: $!\n";

while (<$data>) {
  chomp;

  $sth->execute(split /,/);
}
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Looks far simpler than you were trying make it. Am I missing something? And, yes, it does deal with strings with embedded quotation marks.

Oh, wait. I see what I'm missing. You don't need the quotation marks from the input data. I think you should look at Text::ParseWords.