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');
        }
    });
}