The stream was published as Sucker River by the U.S. Coast and Geodetic Survey (USC&GS) in 1890.

The current name was mentioned by R. G. McConnell of the Geological Survey of Canada in 1888.

The source of the Rapid River is on the slopes of Lahchah Mountain at an elevation of 745 metres (2,444 ft), about 40 kilometres (25 mi) northwest of Old Crow.

The river travels 15 kilometres (9 mi) west to reach the United States border, where it crosses into Yukon–Koyukuk Census Area, Alaska at 67°38′30″N 141°00′00″W / 67.64167°N 141.00000°W / 67.64167; -141.00000 at an elevation of 476 metres (1,562 ft).

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