The following will do, although it redirects to the Posts listing page /wp-admin/edit.php…

Also included, search boxes for Pages and CPTs.

Code

/**
 * PLEASE NOTE THAT THE FOLLOWING CODE DOESN'T HAVE ANY SANITIZATION or NONCE methods
*/

add_action('wp_dashboard_setup', 'wpse_58520_dashboard_search_widget');

function wpse_58520_dashboard_search_widget() {
    wp_add_dashboard_widget( 'wpse_54742_active_site_plugins', __( 'Search Posts/Pages/CPTs' ), 'wpse_58520_make_dashboard_search_widget' );
}

function wpse_58520_make_dashboard_search_widget() {
    ?>
        <form id="posts-filter" action="/wp-admin/edit.php" method="get">
        <div class="search-box" style="height:50px">
            <input type="search" id="post-search-input" name="s" value="">
            <input type="submit" name="" id="search-submit" class="button" value="Search Posts"></p>
        </div>
        </form>

        <form id="pages-filter" action="/wp-admin/edit.php" method="get">
        <div class="search-box" style="height:50px">
            <input type="search" id="page-search-input" name="s" value="">
            <input type="hidden" name="post_type" value="page" /> 
            <input type="submit" name="" id="page-search-submit" class="button" value="Search Pages"></p>
        </div>
        </form>

        <form id="cpts-filter" action="/wp-admin/edit.php" method="get">
        <div class="search-box" style="height:50px">
            <input type="search" id="cpt-search-input" name="s" value="">
            <input type="hidden" name="post_type" value="gallery" /> 
            <input type="submit" name="" id="cpt-search-submit" class="button" value="Search Galleries"></p>
        </div>
        </form>
    <?php
}

Results

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