Assassinate works if the other person has no had a turn in combat yet. Since the first round of combat is a surprise round, this allows you to Assassinate during the Surprise round, and then if your initiative is higher to assassinate somebody else before they can react.
In role-playing terms imagine the Assassin gets the drop on 2 guards. She stabs one with her dagger, and before the other one can turn around and yell out she brings her stiletto up and kicks him in the stomach, before darting forward and slicing his throat. Now, one major flaw here is believing this takes 12 seconds.
Each round may last 6 seconds for ease of use, but that includes enough time to move, last out a few times with weapons and generally perform actions that do require that length of time. But if the Assassin merely drops down, slits the first's throat then kicks the second to prevent a cry of warning, she has sufficient time (even if it's only a couple of seconds) to take him down.
Of course, the logic can break down in that an Assassin who gets the drop on a foe can then move 120' to Assassinate a second one just by beating them in initiative. But feel free to come up with alternatives for Initiative, for example:
1. Variant: Surprise as a bonus to initiative.
Instead of preventing people from acting during a surprise round, simply chose to grant those who are not surprised a bonus to initiative. It could be as low as 10 (meaning there's a good chance they go first, but not always), 20 (excellent chance but still allows really aware people the chance not to suffer) or 100 (surprised folks always go last)
2. Variant: Surprise is not a full round.
Instead of allowing a full round of surprise, simply deny them a move action during a surprise round. This means a Rogue could still cunning action to get into position, but very few other combatants could move and attack. You could deny them bonus actions too if you only want people to have an Action and Bonus Action.
Obviously you could also simply deny an Assassin the opportunity to use Assassinate more than once per encounter, but I don't think going quite so far is necessary for a power level.