Athletic director Eric Zillmer lauded Spiker on Friday for his "unbelievable energy" and calls him a "true CEO and modern coach of college basketball."
Spiker, 39, took over at Army in 2009 and had a 102-112 career mark in seven seasons with the Black Knights. He was previously an assistant coach at Cornell.
Army was 19-14 this season and 9-9 in Patriot League play.
Spiker will replace Bruiser Flint, who was let go earlier in the month after 15 seasons. Drexel has not made the NCAA tournament since 1996.
The Associated Press contributed to this report.