The Ravad, in his commentary to the Sifra (at the beginning of parshas Bechukosai (1:3), explains that the obligation to remember what Amalek did to us that we must fulfill with our mouths is accomplished by learning the halachos that pertain to this mitzvah.
Based on this we can explain the answer that the Acharonim suggested, namely that the berachah on the mitzvah of reading parshas Zachor is the birchas haTorah that we recite in the morning (or before reading the Torah). This is because the mitzvah of remembering what Amalek did to us is fulfilled by learning the halachos that pertain to the mitzvah. The learning of these halachos begins with the reading of the parshah in the Torah that discusses this mitzvah.