The Airflow REST API Plugin would help you out here. Once you have followed the instructions for installing the plugin you would just need to hit the following url: http://{HOST}:{PORT}/admin/rest_api/api/v1.0/trigger_dag?dag_id={dag_id}&run_id={run_id}&conf={url_encoded_json_parameters}, replacing dag_id with the id of your dag, either omitting run_id or specify a unique id, and passing a url encoded json for conf (with any of the parameters you need in the triggered dag).

Here is an example JavaScript function that uses jQuery to call the Airflow api:

function triggerDag(dagId, dagParameters){
    var urlEncodedParameters = encodeURIComponent(dagParameters);
    var dagRunUrl = "http://airflow:8080/admin/rest_api/api/v1.0/trigger_dag?dag_id="+dagId+"&conf="+urlEncodedParameters;
    $.ajax({
        url: dagRunUrl,
        dataType: "json",
        success: function(msg) {
            console.log('Successfully started the dag');
        },
        error: function(e){
           console.log('Failed to start the dag');
        }
    });
}

A new option in airflow is the experimental, but built-in, API endpoint in the more recent builds of 1.7 and 1.8. This allows you to run a REST service on your airflow server to listen to a port and accept cli jobs.

I only have limited experience myself, but I have run test dags with success. Per the docs:

/api/experimental/dags/<DAG_ID>/dag_runs creates a dag_run for a given dag id (POST).

That will schedule an immediate run of whatever dag you want to run. It does still use the scheduler, though, waiting for a heartbeat to see that dag is running and pass tasks to the worker. This is exactly the same behavior as the CLI, though, so I still believe it fits your use-case.

Documentation on how to configure it is available here: https://airflow.apache.org/api.html

There are some simple example clients in the github, too, under airflow/api/clients

You should look at Airflow HTTP Sensor for your needs. You can use this to trigger a dag.