Cage did, enabled by the unique physiology of the Omega.
To elaborate, the film explained that the Omega was the thing that rebooted the time loop. The death of an Alpha was the trigger, but the actual mechanism was the Omega.
Somehow, by killing the Omega, and then dying himself (with the Omega blood mingled into his own), Cage started a new time loop. Something about the Omega's ability to manipulate time meant that when this new time loop started, the Omega's destruction was carried over. I'd argue this is due to the Omega not just remembering past time loops but somehow physically going back in time itself when a time loop restarts. Its destruction therefore brought its dead form back in time with the new time loop. From the standpoint of all the humans, this meant there was a living Omega that suddenly was replaced with a dead Omega, and this triggered some large energy release, as reported in the news at the end of the film.
Since the film takes a perspective of the humans, we have limited information on the aliens and their capabilities. We only know what the humans know, so we don't know for certain what happens normally when an Omega is destroyed.