Other people will probably advice you some module for parsing CSV files.
I solved this problem a year ago, and I found this trick:

sub parseCSVrow {
  my ($row, $fdelim, $tdelim) = @_;
  chomp($row);
  my @ret = split(/$fdelim/,$row);
  my $i = 0;
  while($i < @ret) {
    if (($ret[$i] =~ s/$tdelim/$tdelim/ge) % 2) {
      if ($i + 1 == @ret) {
        die "ERROR: not ending text";
      } else {
        $ret[$i] = $ret[$i].$fdelim.$ret[$i+1];
        splice(@ret,$i+1,1);
      }
    } else {
      $i++;
    }
  }
  return \@ret;  
}
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The idea is to join previously split parts, if some part contains odd count of text delimiters. But, there is one big mistake. CSV field can contain EOL and in this case my code piece does not work.
Update: Respectively, works, but you cannot naively call this subroutine, but slighlty change the code.