This will do:

$ cat replica
my @dialers;
while (<DATA>) {
  chomp;
  push @dialers, $1
    if /replica\s+"([^"]+)/;
}

print $_, "\n" for @dialers;

__DATA__
2005-04-01     root        replica "ml_v_dialer"
2004-06-22     root        replica "pu_v_dialer"
2006-02-11     ccvob01     replica "rd_v_dialer" "v_dialer replica for
+ Redmond"
2003-11-25     root        replica "v_dialer_drcalvin"

$ perl replica
ml_v_dialer
pu_v_dialer
rd_v_dialer
v_dialer_drcalvin
[download]

You can also use split to achieve the same, so instead of,

  push @dialers, $1
    if /replica\s+"([^"]+)/;
[download]

you can write,

  (my $dialer = (split)[3]) =~ s/"//g;
  push @dialers, $dialer;
[download]