Tillman (7-2) yielded only Dayan Viciedo's fifth-inning single en route to the dominating victory, during which he tossed 67 of his 109 pitches for strikes.
The right-hander worked around four walks and struck out five as the American League wild card-contending Orioles grabbed a 2-0 lead in their four-game set against the White Sox.
Brian Matusz completed the whitewash by surrendering one more Chisox hit over two scoreless frames.
Markakis delivered a bases-clearing double in the second inning while Lew Ford and Adam Jones each supplied a solo homer for the victors.
Chris Sale (15-5) took the loss after giving up four runs on six hits over four full innings for AL Central-leading Chicago.
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