The verb describes what you "do" - therefore bringen and halten indicate two different concepts.
Let's use a burning house as an example. Extinguishing a fire is more than randomly spraying water on it. It's a process that can take many hours, during which the firecrew stops the fire's expansion and decreases the burning area until the fire is completely extinguished.
When a house is on fire and the firemen arrive at the scene, their first goal is to make sure the fire does not expand and they can start extinguish the fire in a controlled fashion. They have to gain control, in German
Das Feuer unter Kontrolle bringen.
During the actual process of extinguishing, they decrease the burning area while making sure that no random sparks or unobserved parts allow for re-ignition or even expansion of the fire, they keep it under control, in German
Das Feuer unter Kontrolle halten.
Nehmen is rarely used with Kontrolle, it would typically be übernehmen, which is a switch in responsibility. The full construction is
die Kontrolle (über accusative) (von dative) übernehmen
In your previous example, a second crew can take over after a certain time. It typically involves some kind of hand-over.