Running Job On Airflow Based On Webrequest

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');
        }
    });
}

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