Yes, that'll work. As will

$post{$element} = $1
  if $html =~ m/id="$element" name="challenge" value="(.*?)"/;
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Unless you expect $element to contain metacharacters, you should use

m/id="\Q$element\E" name="challenge" value="(.*?)"/;
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to escape any metacharacters in $html. See quotemeta for more.