With the game tied 1-1 in the fourth, Kila Ka'aihue drew a lead-off walk from Justin Masterson (0-2) and Smith then blasted 1-0 pitch inside the right field foul pole for a two-run shot, putting Oakland ahead for good.
Ross (1-0) limited the Indians to one run on four hits over 6 2/3 innings, overcoming five walks to post the victory.
Jordan Norberto, Ryan Cook and Grant Balfour combined for 2 1/3 frames of scoreless relief and Chris Pennington drove in a pair of runs as the A's salvaged one win in their three-game set against Cleveland.
Masterson yielded four runs on six hits in a five-inning start for the Tribe.
Celebrity Families of 2014 [PHOTOS]