Another solution, this one without any loops:

use strict;
use warnings;

my $data;
{
    local $/ = undef;
    $data = <DATA>;
}

# my @results = $data =~ m/First Line\nSecond Line\n(\d+)\n/mg;
# The m modifier was not really necessary here

my @results = $data =~ m/First Line\nSecond Line\n(\d+)\n/g;
print join "\n", @results;

__DATA__
First Line
Second Line
123
First Line
Second Line
456
First Line
Second Line
789
First Line
Second Line
101112
First Line
Second Line
131415
[download]

Output:

123
456
789
101112
131415
[download]