The Show_HTML sub should return the html, not print it.
It is not mandatory to use a here-doc, you could built it up like this ; 
sub Show_HTML {
  
  my $html = $cgi->start_html();
  $html .= $cgi->p('Enter something');
  $html .= $cgi->input( {-type=>'text', -name=>"va1",-id=>"val1", 
  onkeyup=>"exported_func( ['val1'], ['resultdiv'] );"} );
  
  $html .= $cgi->br;
  $html .= $cgi->div( {-id=>"resultdiv"} );
  $html .= $cgi->end_html();  
  return $html;
}

[download]
poj

In reply to Re: CGI::Ajax No head/html tags, nowhere to insert. Returning javascript anyway. by poj
in thread CGI::Ajax No head/html tags, nowhere to insert. Returning javascript anyway. by msinfo

Title:
Use:  <p> text here (a paragraph) </p>
and:  <code> code here </code>
to format your post, it's "PerlMonks-approved HTML":


  • Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
  • Titles consisting of a single word are discouraged, and in most cases are disallowed outright.
  • Read Where should I post X? if you're not absolutely sure you're posting in the right place.
  • Please read these before you post! —
  • Posts may use any of the Perl Monks Approved HTML tags:
    a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, details, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, summary, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
  • You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)
            For:     Use:
    & &amp;
    < &lt;
    > &gt;
    [ &#91;
    ] &#93;
  • Link using PerlMonks shortcuts! What shortcuts can I use for linking?
  • See Writeup Formatting Tips and other pages linked from there for more info.