The best way to send messages from server to client right now is using webSockets. The basic concept is this:
Server B accepts that webSocket connection and the socket stays open for the duration of the life of the web page.
Server B registers event handlers to handle incoming messages from the web page.
Client A registers event handlers to handle incoming messages from the server.
At any point in time, the server can proactively send data to the client page and it will receive that data.
At any point in time, the client may sent data to the server and it will receive that data.
A popular node.js library that makes webSocket support pretty easy is socket.io. It has both client and server support so you can use the same library for both ends of the connection. The socket.io library supports the .emit() method mentioned in your question for sending a message over an active webSocket connection.
You don't directly call functions from client to server. Instead, you send a message that triggers the server to run some particular code or vice versa. This is cooperative programming where the remote end has to be coded to support what you're asking it to do so you can send it a message and some optional data to go with the message and then it can receive that message and data and execute some code with that.
So, suppose you wanted the server to tell the client anytime a temperature changed so that the client could display in their web page the updated temperature (I actually have a Raspberry Pi node.js server that does exactly this). In this case, the client web page establishes a webSocket connection to the server when the page loads. Meanwhile, the server has its own process that is monitoring temperature changes. When it sees that the temperature has changed some meaningful amount, it sends a temperature change message to each connected client with the new temperature data. The client receives that message and data and then uses that to update it's UI to show the new temperature value.
The transaction could go the other way too. The client could have a matrix of information that it wants the server to carry out some complicated calculation on. It would send a message to the server with the type of calculation indicated in the message type and then send the matrix as the data for the message. The server would receive that message, see that this is a request to do a particular type of calculation on some data, it would then call the appropriate server-side function and pass it the client data. When the result was finished on the server, it would send a message back to the client with the result. The client would receive that result and then do whatever it needed to with the calculated result.
Note, if the transactions are only from client to server with a response then coming back from the server, a webSocket is not needed for that type of transaction. That can be done with just an Ajax call. Client makes ajax call to the server, server formulates a response and returns the response. Where webSockets are most uniquely useful is if you want to initiate the communication from the server and send unsolicited data to the client at a time that the server decides. For that, you need some continuous connection between client and server which is what a webSocket is designed to be.
It appears there may be more to your question about how to communicate from a C# server to your node.js server so it can then notify the client. If this is the case, then since the node.js server is already a web server, I'd just add a route to the node.js server so you can simply do an http request from the C# server to the node.js server to pass some data to the node.js server which it can then use to notify the appropriate client via the above-described webSocket connection. Depending upon your security needs, you may want to implement some level of security so that the http request can only be sent locally from your C# server, not from the outside world to your node.js server.
@Marcus - a challenge with this question is that you never really described the whole problem. Are you asking how a C# application can notify the node.js application to send a webSocket message to the client? Is that the question? Because I've fully described how the node.js server would send a message to the client already. So, is the remaining part of the question how the C# application would communicate with an already runniong node.js server to tell it to notify the client? If so, then I'd suggest a simple Ajax call (e.g. http request) from C# to the node.js server.