And that works too! Here’s how you do it:
Go to the Sleep tab, where you usually set your alarm.
Swipe to the left. Swipe twice. You’ll arrive at a screen where you can start a sleep session with no alarm. If you press "Start" here, Sleep Cycle only listens and analyzes your sleep, but won’t wake you up.
As you wake up on your own, stop the Sleep Cycle app, as you normally would. You’ll get the night’s analysis, without the alarm waking you up.