I needed to do this recently and after trying a few different permutations (using onfocus, onclick of textbox etc), I finally settled on this...
 <input id="autocomplete" name="autocomplete" type="text" 
 value="Choose Document">
 <p>
 <button type="submit" value="Submit" name="submit" id="submit" >
  Submit
 </button>
 </p>
<script type="text/javascript">
$("#autocomplete").autocomplete(
{
source: '@Url.Content("~/Document/DocumentTypeAutoComplete/")' //<--ddl source
, minLength: 0 // <-- This is necessary to get the search going on blank input
, delay: 0
, select: function (event, ui) 
  {
  if (ui.item) {
     $("#autocomplete").val(ui.item.value);//<-- Place selection in the textbox
          $("docForm").submit(); 
          loadDocTypeCreatePartial(ui.item);
          $("#submit").focus(); //This stops the drop down list from reopening 
                                // after an item in the select box is chosen
                                // You can place the focus anywhere you'd like,
                                // I just chose my *submit* button
                }
   }
  }).focus(function () {
    // following line will autoselect textbox text
    // making it easier for the user to overwrite whats in the 
    // textbox
    $(this).select();
    //The below line triggers the search function on an empty string
    $(this).data("autocomplete").search('');
   });
 </script>
This opens the autocomplete ddl list on focus (Even if you have default text in your input box like I do above). 
It also auto-selects the text in the text box to prevent the user from having to clear out the text.
Once an item is selected from the auto-complete list, it puts that item into the auto-complete input box and moves the focus to another control (thus preventing the auto-complete from reopening).
I plan on replacing it by adding dynamic Ajax calls to the very nice Chosen select lists with the Melting Ice upgrade when I get a chance.