Using queues to handle asynchronous callbacks

The application I'm working on has to fire off multiple requests to an external API. It will receive the results of those requests asynchronously, but then I need to handle each response in a synchronous way. My research tells me there are multiple ways of doing this, but I am considering this idea and wondering if it will work:

Every time I make a call to the external API, its response is entered into the queue as soon as it asynchronously comes back:

function genericFetchFunction(url, callback) {
    makeAsyncRequest(url, (result) => {
        addToHandlingQueue({
            data: result,
            callback
        });
    });
}

The queue handler pushes the response onto the queue, and initiates the queue firing process.

const responseQueue = [];

function addToHandlingQueue(response) {
    responseQueue.push(response);
    if (responseQueue.length == 1) {
        fireQueue();
    }
}

function fireQueue() {
    let item = responseQueue.shift();
    item.callback(item.data);
    if (responseQueue.length > 0) {
        fireQueue();
    }
}

Would this code work how I expect? Will all results get into the queue and their callbacks fired in sequence? If not - why not?