Basically you specify the required cipher explicitly as an option. Now which one to specify, you figure this out by checking the ciphers on server.
Okay I was able to fix this. So basically you understand that curl does not support even one of the ciphers that server is willing to negotiate.
Note that the I was testing this on a ubuntu machine. curl that was present on my ubuntu machine was not using openssl. I think that the default curl that you get by doing sudo apt-get is not built with openssl. So I built curl following this post http://askubuntu.com/questions/764443/how-to-compile-curl-with-ssl-support
After that I was able to make curl request.
I also have a mac and I was able to make curl request successfully from there.
I also had another colleague of mine who was not able to do so from mac.
So I think a solution would be to build curl from source specifying an SSL library to build with ( this may depend on the os that you are using).
Let me know how it goes.