According to Modern Language Association, there is a designated format for citing a tweet right now in an academic paper.
Begin the entry in the works-cited list with the author's real name and, in parentheses, user name, if both are known and they differ. If only the user name is known, give it alone.

Next provide the entire text of the tweet in quotation marks, without changing the capitalization. Conclude the entry with the date and time of the message and the medium of publication (Tweet).
Twitter referencing. Haven't thought about that yet have you?

[Modern Language Association via ScienceRoll]