My players elected to use Disengage in order to bypass Monsters (and avoid OAs) standing between them and their goal. Is this a correct usage of the action? They began their movement outside of Melee Range.
Disengage is the correct action to use in this case. From the basic rules:
If you take the Disengage action, your movement doesn't provoke opportunity attacks for the rest of the turn.
Note that this doesn't place any restrictions on having to be in melee range of a creature to use it.
Dodge would help somewhat, but it only gives those Opportunity Attacks Disadvantage, where Disengage stops them entirely.