Hi vthomeschoolmom
The implementation for Chrome and IE differ somewhat, but you could look at using UIAObjects. I have implemented this previously for downloading files in IE following a similar question finding downloaded files:
https://community.smartbear.com/t5/TestComplete-Functional-Web/determine-file-name-of-downloaded-file/m-p/184958#M34453
https://support.smartbear.com/testcomplete/docs/reference/test-objects/members/window-and-process/uiaobject-method.html
Redacted code:
var frameObj = Aliases.browser.BrowserWindow.WaitWindow("Frame Notification Bar");
var UIANotificationObj = frameObj.UIAObject("Notification");
var objectName = "Save"; //Or whatever the button you are looking to click
var UIAfindObj = UIANotificationObj.UIAObject(objectName);
UIAfindObj.Click();
Of course you may need to map one or two objects. And definately add some waits, checks, retries, refreshmappinginfo(s) to make sure it doesn't fall over, but basically you wait for the notification bar with the save/open/save as buttons and click it
Also, if you know exactly what your file is going to be called, you could look at:
https://support.smartbear.com/viewarticle/8999/
Haven't tried this myself though...