Wordpress - WP_Query with "post_title LIKE 'something%'"?

I would solve this with a filter on WP_Query. One that detects an extra query variable and uses that as the prefix of the title.

add_filter( 'posts_where', 'wpse18703_posts_where', 10, 2 );
function wpse18703_posts_where( $where, &$wp_query )
{
    global $wpdb;
    if ( $wpse18703_title = $wp_query->get( 'wpse18703_title' ) ) {
        $where .= ' AND ' . $wpdb->posts . '.post_title LIKE \'' . esc_sql( $wpdb->esc_like( $wpse18703_title ) ) . '%\'';
    }
    return $where;
}

This way you can still call WP_Query, you just pass the title as the wpse18703_title argument (or change the name to something shorter).

Simplified:

function title_filter( $where, &$wp_query )
{
    global $wpdb;
    // 2. pull the custom query in here:
    if ( $search_term = $wp_query->get( 'search_prod_title' ) ) {
        $where .= ' AND ' . $wpdb->posts . '.post_title LIKE \'%' . esc_sql( like_escape( $search_term ) ) . '%\'';
    }
    return $where;
}

$args = array(
    'post_type' => 'product',
    'posts_per_page' => $page_size,
    'paged' => $page,
    // 1. define a custom query var here to pass your term through:
    'search_prod_title' => $search_term,
    'post_status' => 'publish',
    'orderby'     => 'title', 
    'order'       => 'ASC'
);

add_filter( 'posts_where', 'title_filter', 10, 2 );
$wp_query = new WP_Query($args);
remove_filter( 'posts_where', 'title_filter', 10 );
return $wp_query;

Wanted to update this code you guys worked on for the wordpress 4.0 and above as esc_sql() is deprecated in 4.0 higher.

function title_filter($where, &$wp_query){
    global $wpdb;

    if($search_term = $wp_query->get( 'search_prod_title' )){
        /*using the esc_like() in here instead of other esc_sql()*/
        $search_term = $wpdb->esc_like($search_term);
        $search_term = ' \'%' . $search_term . '%\'';
        $where .= ' AND ' . $wpdb->posts . '.post_title LIKE '.$search_term;
    }

    return $where;
}

Rest of the stuff is same.

Also I want to point out you can use s variable within WP_Query arguments to pass search terms, which will also search for post title i believe.

Like this:

$args = array(
    'post_type' => 'post',
    's' => $search_term,
    'post_status' => 'publish',
    'orderby'     => 'title', 
    'order'       => 'ASC'        
);
$wp_query = new WP_Query($args);