pac-input being your place search input, you can do the following:
// Trigger search on blur
document.getElementById('pac-input').onblur = function () {
    google.maps.event.trigger(this, 'focus', {});
    google.maps.event.trigger(this, 'keydown', {
        keyCode: 13
    });
};
Or with a google maps event listener:
// Trigger search on blur
google.maps.event.addDomListener(document.getElementById("pac-input"), 'blur', function() {
    google.maps.event.trigger(this, 'focus', {});
    google.maps.event.trigger(this, 'keydown', {
        keyCode: 13
    });
});
Edit: As mentioned in the comments, this messes up with the "normal behavior" of a user selecting a place by mouse click.
Below is a complete example using pure Javascript. For a (simpler) jQuery solution, look at @ChrisSnyder's answer.
function initialize() {
  var ac = new google.maps.places.Autocomplete(
    (document.getElementById('autocomplete')), {
      types: ['geocode']
    });
  ac.addListener('place_changed', function() {
    var place = ac.getPlace();
    if (!place.geometry) {
      // User entered the name of a Place that was not suggested and
      // pressed the Enter key, or the Place Details request failed.
      console.log("No details available for input: '" + place.name + "'");
      return;
    }
    console.log("You selected: '" + place.formatted_address + "'");
  });
  // Trigger search on blur
  google.maps.event.addDomListener(document.getElementById("autocomplete"), 'blur', function() {
    // Find the pac-container element
    var pacContainer = nextByClass(this, 'pac-container');
    // Check if we are hovering on one of the pac-items
    // :hover propagates to parent element so we only need to check it on the pac-container
    // We trigger the focus and keydown only if :hover is false otherwise it will interfere with a place selection via mouse click
    if (pacContainer.matches(':hover') === false) {
      google.maps.event.trigger(this, 'focus', {});
      google.maps.event.trigger(this, 'keydown', {
        keyCode: 13
      });
    }
  });
}
function hasClass(elem, cls) {
  var str = " " + elem.className + " ";
  var testCls = " " + cls + " ";
  return (str.indexOf(testCls) != -1);
}
function nextByClass(node, cls) {
  while (node = node.nextSibling) {
    if (hasClass(node, cls)) {
      return node;
    }
  }
  return null;
}
#autocomplete {
  width: 300px;
}
<script src="https://maps.googleapis.com/maps/api/js?libraries=places&callback=initialize&key=AIzaSyCkUOdZ5y7hMm0yrcCQoCvLwzdM6M8s5qk" async defer></script>
<input id="autocomplete" placeholder="Enter your address" type="text"></input>