"MySQL's way of escaping a quote is to double it."

Actually, that's not something specific to MySQL; it's part of the SQL standard.

davido's answer is a good one. In the spirit of "there's more than one way to do it"...

$dbh->do(<<'GO');
UPDATE table_name
SET `fieldname`=REPLACE(`fieldname`,'‘','''')
GO
[download]

... or ...

$dbh->do("UPDATE table_name SET `fieldname`=REPLACE(`fieldname`,'‘',''
+'')");
[download]

... or ...

$dbh->do('UPDATE table_name SET `fieldname`=REPLACE(`fieldname`,\'‘\',
+\'\'\'\')');
[download]

... or ...

$dbh->do(q(UPDATE table_name SET `fieldname`=REPLACE(`fieldname`,'‘','
+''')));
[download]

Lastly, if you've got non-ASCII characters in your source code (such as your "smart quote"), make sure your editor is set up to save your script as UTF-8; and make sure you include the pragma use utf8 somewhere near the top of your script.