iOS 7+ Swift
Swift 4:
// 2018.10.30 par:
//   I've updated this answer with an asynchronous dispatch to the main queue
//   when we're called without animation. This really should have been in the
//   previous solutions I gave but I forgot to add it.
extension UINavigationController {
    public func pushViewController(
        _ viewController: UIViewController,
        animated: Bool,
        completion: @escaping () -> Void)
    {
        pushViewController(viewController, animated: animated)
        guard animated, let coordinator = transitionCoordinator else {
            DispatchQueue.main.async { completion() }
            return
        }
        coordinator.animate(alongsideTransition: nil) { _ in completion() }
    }
    func popViewController(
        animated: Bool,
        completion: @escaping () -> Void)
    {
        popViewController(animated: animated)
        guard animated, let coordinator = transitionCoordinator else {
            DispatchQueue.main.async { completion() }
            return
        }
        coordinator.animate(alongsideTransition: nil) { _ in completion() }
    }
}
EDIT: I've added a Swift 3 version of my original answer. In this version I've removed the example co-animation shown in the Swift 2 version as it seems to have confused a lot of people.
Swift 3:
import UIKit
// Swift 3 version, no co-animation (alongsideTransition parameter is nil)
extension UINavigationController {
    public func pushViewController(
        _ viewController: UIViewController,
        animated: Bool,
        completion: @escaping (Void) -> Void)
    {
        pushViewController(viewController, animated: animated)
        guard animated, let coordinator = transitionCoordinator else {
            completion()
            return
        }
        coordinator.animate(alongsideTransition: nil) { _ in completion() }
    }
}
Swift 2:
import UIKit
// Swift 2 Version, shows example co-animation (status bar update)
extension UINavigationController {
    public func pushViewController(
        viewController: UIViewController,
        animated: Bool,
        completion: Void -> Void)
    {
        pushViewController(viewController, animated: animated)
        guard animated, let coordinator = transitionCoordinator() else {
            completion()
            return
        }
        coordinator.animateAlongsideTransition(
            // pass nil here or do something animated if you'd like, e.g.:
            { context in
                viewController.setNeedsStatusBarAppearanceUpdate()
            },
            completion: { context in
                completion()
            }
        )
    }
}