Is expiation obligatory upon one who swore falsely, saying, "By Allah, I believe," to evade a situation, then realized within seconds that he had sworn falsely, but neither sought forgiveness nor informed the other party, nor fulfilled what he had sworn upon?

If the oath was unintentional, it is a 'laghw' oath (unintended/vain oath) and requires no expiation. However, if it was intentional, and the questioner did not sleep, then expiation is due, because it is a binding oath concerning a future action. She is not obligated to inform the person that she did not sleep, but she must offer expiation if she intended the oath. She could have used 'ma'areedh' (ambiguous statements) to avoid the situation without lying.