The Rapids (0-3-3) had jumped off to a 2-0 lead on goals by Dillion Powers in the 18th minute and Hendry Thomas in the 48th.
Johnson, however, began the comeback in the 55th when he headed home a cross from Ryan Johnson, halving the Timbers' deficit.
Then in the 71st minute, Colorado's Atiba Harris hit David Horst of Portland in his own penalty box, producing a penalty kick that Johnson buried into the upper left corner, earning a point for the Timbers (0-1-3).
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