Tomomi Inada issued the order, public broadcaster NHK said, without mentioning any indication that Pyongyang is preparing to launch such a missile.
Last week Prime Minister Shinzo Abe picked Inada, a close confidante with staunchly nationalist views, as the new defence minister.
NHK said Inada is expected to renew the readiness order every three months so that Tokyo can seamlessly maintain a state of alert.
A Japanese defence ministry spokesman declined to confirm the report.
The US military said the North had actually launched two Rodong intermediate-range missiles simultaneously, but one appeared to have exploded on take-off.
The launches followed a North Korean threat of "physical action" over the planned deployment of a sophisticated US anti-missile system in South Korea, and came just weeks before the start of large-scale joint South Korea-US military exercises.
Pyongyang has conducted a series of missile tests this year in defiance of UN sanctions imposed after its fourth nuclear test in January.
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