Even faster than index() or a regex -- I would imagine -- would be substr() and eq together.

When imagine when you can Benchmark?

use Benchmark 'cmpthese';
my $str1 = 'foobarbazqux';
my $str2 = 'foobar';
my $str2len = length $str2;

cmpthese (3e6, {
  regex   => sub { $str1 =~ /\Q$str2/; },
  regex2  => sub { $str1 =~ /\Q$str2/o; },  ## probably useless test, 
+see japhy's comment below
  index   => sub { index $str1, $str2; },
  substr  => sub { $str2 eq substr $str1, 0, length $str2; },
  substr2 => sub { $str2 eq substr $str1, 0, $str2len; },
});
__END__
             Rate   regex  regex2  substr substr2   index
regex   1063830/s      --    -13%    -25%    -43%    -72%
regex2  1229508/s     16%      --    -14%    -34%    -68%
substr  1421801/s     34%     16%      --    -23%    -63%
substr2 1851852/s     74%     51%     30%      --    -51%
index   3797468/s    257%    209%    167%    105%      --
[download]

The index solution is clearly superior to the others. It's interesting to see the improvements of regex2 and substr2 over their unoptimized counterparts.

Update: I forgot that the OP was matching at the beginning of the string. This is a better benchmark:

use Benchmark 'cmpthese';
my $str1 = 'foobarbazqux';
my $str2 = 'foobar';
my $str2len = length $str2;

cmpthese (3e6, {
  regex   => sub { $str1 =~ /^\Q$str2/; },
  index   => sub { 0 == index $str1, $str2; },
  substr  => sub { $str2 eq substr $str1, 0, length $str2; },
  substr2 => sub { $str2 eq substr $str1, 0, $str2len; },
});
__END__
             Rate   regex  substr substr2   index
regex    671141/s      --    -59%    -65%    -71%
substr  1630435/s    143%      --    -16%    -30%
substr2 1935484/s    188%     19%      --    -17%
index   2343750/s    249%     44%     21%      --
[download]