Exultantly tells you how someone did something: with great joy and pride, the kind that comes after a win. “She smiled exultantly” means the smile is not just happy — it is the smile of someone who has come out on top. It is a formal adverb, mostly at home in writing. In talk, people more often say happily, proudly, or with a huge grin. Though actually, because it carries that “I won” flavour, it can sound a little sharp if the success hurt someone else. Use it when the feeling is big, public, and tied to a result.