Thanks. I forgot to include the

memoize('lcs');

that's why my code took so long to run. I also found this brute force method in Perl Monk and ran a comparison on both method and found that the brute force method actually ran faster. This does not make sense at all.
LCSS method:

# lcs.pl
use strict;
use Memoize;

sub longerOf {
    my ($x, $y) = @_;
    return (length $x > length $y) ? $x : $y;
}

memoize('lcs');
sub lcs {
    my ($a, $b) = @_;

    if ($a eq "" || $b eq ""){
        return "";
    }

    my ($az, $bz) = (chop $a, chop $b);
    if ($az eq $bz){
        return lcs($a, $b) . $az;
    } else {
        return longerOf(
                  lcs($a . $az, $b),
                  lcs($b . $bz, $a));
    }
}

while (1){
    print "1: "; my $a = <>; chomp $a;
    print "2: "; my $b = <>; chomp $b;
$start = time();
    print "LCS: ", lcs($a, $b), "\n\n";
  $end = time();
   print "<br>Time taken was ", ($end - $start), " seconds";

$start = time();
    print "Brute Force: ", lcsbruteforce($a, $b), "\n\n";
  $end = time();
   print "<br>Time taken was ", ($end - $start), " seconds";
}


sub lcsbruteforce {
  my($x, $y) = @_;
  my(@v, $cx, $cy, $left, $above);
  for my $xi (0 .. length($x) - 1) {
    $cx = substr $x, $xi, 1;
    for my $yi (0 .. length($y) - 1) {
      $cy = substr $y, $yi, 1;
      if ($cx eq $cy) {
        $v[$xi][$yi] = 1 + (($xi && $yi) ? $v[$xi - 1][$yi - 1] : 0);
      } else {
        $left = ($xi && $v[$xi - 1][$yi]) || 0;
        $above = ($xi && $v[$xi][$yi - 1]) || 0;
        $v[$xi][$yi] = ($left > $above) ? $left : $above;
      }
    }
  }
  return $v[length($x) - 1][length($y) - 1];
}
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