Following on from the previous replies here is a benchmark demonstrating the performance difference. Note though that with the test string the speed difference is only of the order of two times, not the 10 times described by OP.

use warnings;
use strict;
use Benchmark qw(cmpthese);

my $target = 'This is a string used to test the time required for a gr
+eedy match compared to a non-greedy match.';
my $greedy = qr/(\ba\b.*\bstring\b)/;
my $non = qr/(\ba\b.*?\bstring\b)/;

my ($matchG) = $target =~ $greedy;
my ($matchN) = $target =~ $non;

die "Matches generate different results\n" if $matchG ne $matchN;

cmpthese
  (
  -1,
    {
    'Greedy' => sub {$target =~ $greedy;},
    'Non' => sub {$target =~ $non;}
    }
  );

Prints:

           Rate Greedy    Non
Greedy 162689/s     --   -64%
Non    456847/s   181%     --
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