2020 Update (for Woocommerce 3+)

Replaced deprecated function woocommerce_price() by wc_price() since Woocommerce 3+:

add_filter('woocommerce_variable_sale_price_html', 'shop_variable_product_price', 10, 2);
add_filter('woocommerce_variable_price_html','shop_variable_product_price', 10, 2 );
function shop_variable_product_price( $price, $product ){
    $variation_min_reg_price = $product->get_variation_regular_price('min', true);
    $variation_min_sale_price = $product->get_variation_sale_price('min', true);
    if ( $product->is_on_sale() && !empty($variation_min_sale_price)){
        if ( !empty($variation_min_sale_price) )
            $price = '<del class="strike">' .  wc_price($variation_min_reg_price) . '</del>
        <ins class="highlight">' .  wc_price($variation_min_sale_price) . '</ins>';
    } else {
        if(!empty($variation_min_reg_price))
            $price = '<ins class="highlight">'.wc_price( $variation_min_reg_price ).'</ins>';
        else
            $price = '<ins class="highlight">'.wc_price( $product->regular_price ).'</ins>';
    }
    return $price;
}

Original thread (before WooCommerce 3): The code below should do what you expect:

add_filter('woocommerce_variable_sale_price_html', 'shop_variable_product_price', 10, 2);
add_filter('woocommerce_variable_price_html','shop_variable_product_price', 10, 2 );
function shop_variable_product_price( $price, $product ){
    $variation_min_reg_price = $product->get_variation_regular_price('min', true);
    $variation_min_sale_price = $product->get_variation_sale_price('min', true);
    if ( $product->is_on_sale() && !empty($variation_min_sale_price)){
        if ( !empty($variation_min_sale_price) )
            $price = '<del class="strike">' .  woocommerce_price($variation_min_reg_price) . '</del>
        <ins class="highlight">' .  woocommerce_price($variation_min_sale_price) . '</ins>';
    } else {
        if(!empty($variation_min_reg_price))
            $price = '<ins class="highlight">'.woocommerce_price( $variation_min_reg_price ).'</ins>';
        else
            $price = '<ins class="highlight">'.woocommerce_price( $product->regular_price ).'</ins>';
    }
    return $price;
}

The Code goes in function.php file of your active child theme (or theme) or also in any plugin file.

This code is tested and works.