<dependency>
  <groupId>org.apache.httpcomponents</groupId>
  <artifactId>httpclient</artifactId>
  <version>4.5.3</version>
</dependency>

private Cookie login(URI targetUri, String loginPath, Map<String, String> params) throws IOException {
    requireNonNull(targetUri);
    requireNonNull(loginPath);

    // Keep track of cookies we might receive in an HttpClient:
    final CookieStore cookies = new BasicCookieStore();

    // Build and work with an (AutoCloseable) HttpClient that uses the cookie store:
    try (CloseableHttpClient client = HttpClients.custom().setDefaultCookieStore(cookies).build()) {

        // Prepare (login) request parameters:
        List<NameValuePair> reqParams = new ArrayList<>();
        if (params != null) {
            for (Map.Entry<String, String> entry : params.entrySet()) {
                reqParams.add(new BasicNameValuePair(entry.getKey(), entry.getValue()));
            }
        }

        // Execute the login (POST) request with the given parameters:
        HttpPost post = new HttpPost(targetUri + loginPath);
        post.setEntity(new UrlEncodedFormEntity(reqParams));

        CloseableHttpResponse response = client.execute(post);
        // Eventually, check the response to see if successful
        response.close();

        // Look for a JSESSIONID-named cookie stored in the HttpClient and return it to be used by calling code:
        for (org.apache.http.cookie.Cookie cookie : cookies.getCookies()) {
            if ("JSESSIONID".equalsIgnoreCase(cookie.getName())) {

                String domain = targetUri.getHost();
                if (domain.startsWith("www.")) {
                    domain = domain.substring(4);
                }

                Cookie authCookie = new Cookie(cookie.getName(), cookie.getValue());
                authCookie.setDomain(domain);
                authCookie.setPath("/");
                // Eventually, set expiry (to allow longer login) and other things...

                return authCookie;
            }
        }

        return null; // some sort of error?
    }
}

@Title("Backoffice for SO Question #42927030")
public class MainUI extends UI {

    private Cookie login(URI targetUri, String loginPath, Map<String, String> params) throws IOException {
        // ...
    }

    @Override
    protected void init(VaadinRequest vaadinRequest) {
        setContent(new VerticalLayout(new Button("Log into site...", event -> {

            try {
                URI targetUri = new URI("http://localhost:8080");

                Map<String, String> params = new HashMap<>();
                params.put("username", "Johnny");
                params.put("password", "incorrect :)");
                // Eventual hidden fields, etc.
                // params.put("...", "...");

                Cookie targetAuthCookie = login(targetUri, "/log-me-in", params);

                // We're not ready just yet: we still need to 'transfer' the cookie
                // the HTTP client received to the current browser:
                VaadinService.getCurrentResponse().addCookie(targetAuthCookie);

                // Upon responding to the Vaadin 'click' request, open the target URL (eventually in a new page / tab):
                Page.getCurrent().getJavaScript().execute("window.open('" + targetUri + "');");

            } catch (Exception ex) {
                ex.printStackTrace();
            }

        })));
    }

    @WebServlet(urlPatterns = "/*", name = "MainUIServlet", asyncSupported = true)
    @VaadinServletConfiguration(ui = MainUI.class, productionMode = false)
    public static class MainUIServlet extends VaadinServlet {}
}