Holtby's third career whitewash backstopped a goal and an assist from Alex Ovechkin as the Capitals grabbed sole possession of the eighth and final Eastern Conference playoff berth.
At 84 points, the Capitals are tied with Buffalo but hold the edge over the Sabres on tiebreakers.
Jason Chimera and Mathieu Perreault also scored for Washington, which shook off four losses in its previous five games to record the much-needed win.
Matt Hackett was saddled with the loss after yielding three goals on 23 shots for the Wild, who were officially eliminated from post-season contention with their fifth loss in seven games.
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