One of the most nefarious villains introduced in Dragon Ball Super is Zamasu, a god who trained to become the Supreme Kai of Universe 10 before deciding to seize Goku's body in a divergent timeline to wipe out all mortals in existence. As the threat of Zamasu -- subsequently named Goku Black -- loomed over multiple timelines within Universe 7, Lord Beerus thought he quickly solved the problem by erasing the Zamasu from the prime Universe 10 timeline. The God of Destruction was confident this strategy would similarly erase Future Zamasu. Instead, the alternate future's villain survived to continue to battle Goku and the Z Fighters. But how?
The key to Future Zamasu's survival lies in how he was able to travel to these divergent timelines throughout the Dragon Ball multiverse: The Time Ring. Stored in a special chamber overseen by Gowasu, the Supreme Kai of Universe 10 and Zamasu's mentor, the Time Rings were described by Gowasu as having been created when a human from Universe 12 developed the means to travel through time, with each Time Ring formed by the creation of each divergent timeline. The Rings allowed users to travel across various timelines, with Zamasu seizing one for himself to enact his plans after murdering Gowasu in the future timeline.
The incident with Goku Black may be one of the first major instances of time travel and its repercussions that Beerus faced in recent history -- Gowasu observes that the last time a Time Ring was used was approximately 400 years before he began training Zamasu. Given that the Time Rings are kept with the Supreme Kai of an alternate universe, Beerus' status as the God of Destruction of Universe 7 may have left him unsure of how the mechanics of the Time Rings worked, including that they apparently protect those that wear them from the effects of creating temporal paradoxes, as Beerus had done by erasing the prime timeline's Zamasu.
The other major factor in Future Zamasu's survival is that the time travel rules Beerus assumed existed have never been particularly evident throughout Dragon Ball. When Future Trunks traveled to the prime timeline from his alternate future to destroy the Androids during Dragon Ball Z, his victory did not change the events that took place in his own timeline, which was still menaced by the Androids who plagued him all his life. This avoids a temporal paradox from taking place by simply creating a divergent timeline, further evidenced by each Time Ring forming as different timelines are formed by those traveling through time. This is a time travel mechanic that Future Trunks is well aware of but Beerus may not be overly familiar with.
Beerus has always been perhaps overconfident, convinced of his own infallibility and offended at any insinuation that he may not be as supremely powerful or all-knowing as he presents himself in front of mortals to be. Not used to having to kill an opponent twice, Beerus' showdown against the prime timeline's Zamasu in Dragon Ball Super is likely far outside the realm of his normal experiences and operational knowledge. Beerus was successful in defeating a Zamasu but certainly not all Zamasus, proving that the conflict against Zamasu and Black Goku would not be solved anywhere near as easily as Beerus believed. And as Zamasu not only survived but grew more powerful, it quickly became apparent that it would take an Omni King instead of a God of Destruction to save the day.