It's important to anchor both ends of a non-greedy match because the minimum that can be matched is nothing at all.

use strict;
use warnings;

my $str = "Some rubbish before TEST and the good stuff afterwards";

$str =~ s/^.*?(?=TEST)//;

print $str;
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Prints:

TEST and the good stuff afterwards
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Not too the use of (?=TEST) which is a zero width look ahead match. It matches TEST, but doesn't consume it so you don't need to replace it later.