Tillman (10-2) registered his sixth win in six starts, posting a 2.68 ERA over the span. The right-hander limited New York to two runs on five hits.
With the sweep, the second-place O's moved to 2 1/2 games behind Boston in the American League East.
Yankees starter Hiroki Kuroda (7-6) surrendered four runs on seven hits over six full frames.
Robinson Cano homered for slumping New York, which dropped a fifth straight and fell 6 1/2 games off the pace in the East.
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