Jquery When and Deferred object, broken function flow

For me the code is overly divided into functions, with some doing little more than fronting for others.

I would prefer to see the click handler as a comprehensive master routine which sequences three promise-returning functions zSaveMasterData(), zSaveDetails() and zAuditSave(), then closes the window. Thus, some of the current functions will be subsumed by the click handler.

$("#btnSaveAndClose").click(function(event) {
    zSaveMasterData(masterdata).then(function() {
        var selectedDataGroups;
        /* some logic here */
        var detailsSaved = zSaveDetails(selectedDataGroups).then(function(data) {
            var printableGroupIDs = $.map(data, function (obj) {
                return obj.IsPrintable ? obj.ID : null;
            });
            if (printableGroupIDs.length > 0) {
                // calls external program to print groups
            }
        });
        // Here, it is assumed that zSaveDetails() and zAuditSave() can be performed in parallel.
        // If the calls need to be sequential, then the code will be slightly different.            
        return $.when(detailsSaved, zAuditSave());
    }).then(function() {
        window.close();
    });
});

function zSaveMasterData(masterdata) {
    return $.ajax({
        type: 'POST',
        url: '/api/masterdata/',
        contentType: 'application/json',
        data: JSON.stringify(masterdata),
    });
};

function zSaveDetails(selectedDataGroups) {
    return $.ajax({
        type: 'POST',
        contentType: 'application/json',
        url: '/api/detaildata/',
        data: JSON.stringify(selectedDataGroups)
    });
};

function zAuditSave() {
    return $.ajax({
        type: 'POST',
        contentType: 'application/json',
        url: '/api/audit'
    });
};

Note the returns in the three functions with ajax calls. These returns are vital to the sequencing process.

A potentially bigger issue, not addressed in the question (nor in this answer) is how to recover from errors. Presumably, the database will be inconsistent if the sequence of saves was to fail part way through. It may well be better to ditch this client-side sequencing approach in favour of a server-side transaction that the client sees as a single operation.

The problem here is your code doesn't depend on when fnAfterSave() has completed.

Short answer: don't mix success methods, callbacks, and promises - use one pattern and stick to it - and the easiest pattern to use is promises.

$("#btnSaveAndClose").click(function (event) {
    zSaveSomeData().then(function() { zCloseMyWindow(); });
});

function zSaveSomeData() {
    return zSaveMasterData(masterdata).then(function(data) { zSaveDetailData() });
};

function zSaveMasterData(masterdata) {
    return $.ajax({
        type: 'POST',
        contentType: 'application/json',
        url: '/api/masterdata/',
        data: JSON.stringify(masterdata)
    });

    //remove success callback here as it breaks the chaining
};

It seems like your problem is that you are doing asynchronous things inside an ajax success callback. The promise returned by $.ajax still resolves immediately after the response is received - and executes your done callback before the asynchronous zSaveDetailData() has finished.

So, to chain asynchronous actions, always use then. Use it even for synchronous actions, it makes the sequence clear.

Don't use success callbacks when you're working with promises. You also don't need deferreds. You might want to have a look at these generic rules as well, especially that you never must forget to return promises from async functions that you want to await.

$("#btnSaveAndClose").click(function (event) {
    zSaveSomeData().then(zCloseMyWindow);
});

function zSaveSomeData() {
    return zSaveMasterData(masterdata).then(zSaveDetailData);
}

function zSaveMasterData(masterdata) {
    return $.ajax({
        type: 'POST',
        contentType: 'application/json',
        url: '/api/masterdata/',
        data: JSON.stringify(masterdata),
    });
}

function zSaveDetailData() {
    var selectedDataGroups;
    // some logic here

    return zSaveDetails(selectedDataGroups);
//  ^^^^^^
}

function zSaveOrderGroups(selectedDataGroups) {
    return $.ajax({
//  ^^^^^^
        type: 'POST',
        contentType: 'application/json',
        url: '/api/detaildata/',
        data: JSON.stringify(selectedDataGroups)
    }).then(function(data) {
//    ^^^^^^^^^^^^^^^^^^^^^^
        var printableGroupIDs = [];
        $.each(data, function () {
            if (this.IsPrintable)
                 printableGroupIDs.push(this.ID);
        });
        if (printableGroupIDs.length > 0) {
            return zPrintGroups(printableGroupIDs);
//          ^^^^^^
        }
    }).then(zAuditSave);
//    ^^^^^^^^^^^^^^^^^
}

function zPrintGroups(newGroupIDs) {
    // calls external program to print groups
}

function zCloseWindow() {
    window.close();
}

function zAuditSave() {
    return $.ajax({
//  ^^^^^^
        type: 'POST',
        contentType: 'application/json',
        url: '/api/audit'
    });
}