Alcaraz is the ninth and youngest person to have won both legs of the Sunshine Double in his career, Indian Wells and Miami.
Medvedev beat his opponent 6-7 (5-7), 7-6 (7-5), 7-5 in what was an absolute classic match between two of the best players in world tennis today.
Draper only committed one double fault, recorded win percentages of 93% and 71% on the first and second serves respectively.
"Two Americans, she definitely has a very bright future, and unfortunately only one of us could win today," Rogers said on court after her win.