The business cocktail look seems to be all about dressing to easily transition from a business meeting to after-work drinks, or from the work day to a holiday party (via Glamour).
To check the box on the business piece of the puzzle, don't wear anything that you wouldn't typically don for a meeting with the boss, such as a very low-cut top or super-short skirt. In other words, don't ignore the "business" in business cocktail.
Meanwhile, a bit of research about how to dress business cocktail reveals that any business-appropriate look can be transformed into a more cocktail look by having some fun with your footwear choice (via Who What Wear). Another way to style yourself for the afterparty is to add some fun jewelry or outfit elements that feature interesting necklines or cuts — again, without being super revealing or tight.
Overall, transforming a look from business to business cocktail can be as simple as switching out your skirt from a more conservative silhouette to something with a bit of movement or texture. Slipping on shoes with a bejeweled element and a statement necklace also accomplishes the same vibe.