There's nothing wrong in using an external find command in a pipe open, especially if you're not concerned about portability across systems.

But then you're using perl to run what is fundamentally a shell script. In particular there's absoultely no need to fork out external grep's with that system.

Also you have:

open (FIND_CMD, "find /home/modules |");
chomp (my @findArr = <FIND_CMD>);
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I recommend:

1. using the three args form of open and lexical filhandles,
2. checking errors,
3. avoiding to slurp in a potentially big list, whereas we have a syntactically sweet enough while loop just to process it one item at a time.

my $pattern = '.pm$';
my $FilePattern = "sub user_method";
my $ResultFilePath = '/tmp/ModuleSearch';

foreach my $File ( @findArr )
{
   next if ( $File !~ m/$pattern/ );
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This will match Foo.apm; you may want to learn about \Q (and \E) in perldoc perlop.

But since you're running an external find anyway, why not let it do the dirty job? (-name '*.pm')

   system ("grep -iHl \"$FilePattern\" \"$File\" >> $ResultFilePath ")
+;
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I wish I had not seen that...

All in all it may have been:

#!/usr/bin/perl

use strict;
use warnings;

open my $find, '-|', 'find /home/modules -name "*.pm"' or
  die "Can't run find(1): $!\n";

$\="\n";
while (<$find>) {
    chomp;
    open my $fh, '<', $_ or
      die "Can't open `$_': $!\n";
    while (my $line=<$fh>) {
        print, last if $line =~ /sub user_method/;
    }
}

__END__
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A compressed version (having a huge pile o' magic in it) of which could be:

#!/usr/bin/perl  -ln

use strict;
use warnings;

BEGIN { @ARGV='find /home/modules -name "*.pm"|' }

local @ARGV=$_;
/sub user_method/ and (print $ARGV), last while <> ;

__END__
[download]

(both untested)