Is expiation required for one who has made a covenant with Allah to pray a missed prayer at a specific time, but then was delayed by three minutes due to forgetfulness?

The prevailing scholarly opinion (jumhur) holds that making up missed prayers is an immediate obligation. The relied-upon position of the four madhhabs is that if a prayer is missed intentionally, its make-up is immediately obligatory, and delaying it for this period of time is impermissible. As for this pledge, it functions like an oath according to many scholars. If you broke it out of forgetfulness, there is no consequence for you, because expiation is not required for one who breaks an oath out of forgetfulness. It appears that you have not broken your oath, as this difference in time is customarily overlooked. Thus, it seems you have not violated your oath, and in any case, expiation is not required of you.