c_rath's answer is mostly right, but you don't need to make the view controller a root view controller. You can in fact trigger a segue between the top view on the navigation stack and any other view controller, even from the App Delegate. For example, to push a storyboard view controller, you could do this:
Swift 3.0 and Later
// Access the storyboard and fetch an instance of the view controller
let storyboard = UIStoryboard(name: "Main", bundle: nil);
let viewController: MainViewController = storyboard.instantiateViewController(withIdentifier: "ViewController") as! MainViewController;
// Then push that view controller onto the navigation stack
let rootViewController = self.window!.rootViewController as! UINavigationController;
rootViewController.pushViewController(viewController, animated: true);
Swift 2.0 and Earlier
// Access the storyboard and fetch an instance of the view controller
var storyboard = UIStoryboard(name: "Main", bundle: nil)
var viewController: MainViewController = storyboard.instantiateViewControllerWithIdentifier("ViewController") as MainViewController
// Then push that view controller onto the navigation stack
var rootViewController = self.window!.rootViewController as UINavigationController
rootViewController.pushViewController(viewController, animated: true)
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