To expand on that, Perl's operator precedence has '=~' higher than '.'.
So to get the '.' notation to work as I want it to, I assume I should (parenthesise) it, like this:
$ perl -e '$a="A";$b="B";print "Matches!\n" if ($a.$b) =~ /A/'
Matches!
$ perl -e '$a="A";$b="B";print "Matches!\n" if ($a.$b) =~ /C/'