If you need HTTP connection, telnet will be enough:
> telnet 10.10.10.10 80
GET /index.html HTTP/1.1
host: important.site.com
In the case of HTTPS you need openssl:
> openssl s_client -connect 10.10.10.10:443 -prexit
GET /index.html HTTP/1.1
host: important.site.com