YOLA, Nigeria—The battle began with two small drones buzzing over a base where more than 500 Nigerian troops guarded the shores of Lake Chad. Then came the clatter of gunfire from a column of armored cars, artillery units and tanks that also blasted jihadist battle songs from mounted speakers.

Within hours, elite forces from one of Africa’s most powerful militaries had abandoned their base and its cache of heavy weapons, routed by an insurgent army fighting under the familiar black and white flag of Islamic State.