Solution 1:

This is how I achieve the effect without requiring a nav-controller. Try this instead:

Swift 4:

import UIKit
class SegueFromLeft: UIStoryboardSegue {
    override func perform() {
        let src = self.source
        let dst = self.destination

        src.view.superview?.insertSubview(dst.view, aboveSubview: src.view)
        dst.view.transform = CGAffineTransform(translationX: -src.view.frame.size.width, y: 0)

        UIView.animate(withDuration: 0.25,
                              delay: 0.0,
                            options: .curveEaseInOut,
                         animations: {
                                dst.view.transform = CGAffineTransform(translationX: 0, y: 0)
                                },
                        completion: { finished in
                                src.present(dst, animated: false, completion: nil)
                                    }
                        )
    }
}

Swift 3:

import UIKit

class SegueFromLeft: UIStoryboardSegue
{
    override func perform()
    {
        let src = self.sourceViewController
        let dst = self.destinationViewController

        src.view.superview?.insertSubview(dst.view, aboveSubview: src.view)
        dst.view.transform = CGAffineTransformMakeTranslation(-src.view.frame.size.width, 0)

        UIView.animateWithDuration(0.25,
            delay: 0.0,
            options: UIViewAnimationOptions.CurveEaseInOut,
            animations: {
                dst.view.transform = CGAffineTransformMakeTranslation(0, 0)
            },
            completion: { finished in
                src.presentViewController(dst, animated: false, completion: nil)
            }
        )
    }
}

Then in the storyboard, click on the segue you'd like to change. In the attributes inspector change the type to 'Custom' and change the class to 'SegueFromLeft'

Solution 2:

Here is a custom-segue class you can use to perform a left-to-right segue in Swift. It requires QuartzCore framework and minimal animation.

Swift 2.2

import UIKit
import QuartzCore

class SegueFromLeft: UIStoryboardSegue {

    override func perform() {
        let src: UIViewController = self.sourceViewController
        let dst: UIViewController = self.destinationViewController
        let transition: CATransition = CATransition()
        let timeFunc : CAMediaTimingFunction = CAMediaTimingFunction(name: kCAMediaTimingFunctionEaseInEaseOut)
        transition.duration = 0.25
        transition.timingFunction = timeFunc
        transition.type = kCATransitionPush
        transition.subtype = kCATransitionFromLeft
        src.navigationController!.view.layer.addAnimation(transition, forKey: kCATransition)
        src.navigationController!.pushViewController(dst, animated: false)
    }

}

Swift 3.0

import UIKit
import QuartzCore

class SegueFromLeft: UIStoryboardSegue {

    override func perform() {
        let src: UIViewController = self.source
        let dst: UIViewController = self.destination
        let transition: CATransition = CATransition()
        let timeFunc : CAMediaTimingFunction = CAMediaTimingFunction(name: kCAMediaTimingFunctionEaseInEaseOut)
        transition.duration = 0.25
        transition.timingFunction = timeFunc
        transition.type = kCATransitionPush
        transition.subtype = kCATransitionFromLeft
        src.navigationController!.view.layer.add(transition, forKey: kCATransition)
        src.navigationController!.pushViewController(dst, animated: false)
    }
}

The "Back" button will still appear in the navigation bar on the transitioned view controller, but you can easily disable/configure the navigation controller for that view controller.

Solution 3:

Updated accepted answer in Swift 3:

import UIKit

class SegueFromLeft: UIStoryboardSegue
{
    override func perform()
    {
        let src = self.source
        let dst = self.destination

        src.view.superview?.insertSubview(dst.view, aboveSubview: src.view)
        dst.view.transform = CGAffineTransform(translationX: -src.view.frame.size.width, y: 0)

        UIView.animate(withDuration: 0.25,
            delay: 0.0,
            options: UIViewAnimationOptions.curveEaseInOut,
            animations: {
                dst.view.transform = CGAffineTransform(translationX: 0, y: 0)
            },
            completion: { finished in
                src.present(dst, animated: false, completion: nil)
            }
        )
    }
}