Rodriguez (1-0) walked one and struck out two. His only mistake was a two-out homer to Tony Batista in the second inning.
Jose Santiago gave up a leadoff homer to Deivi Cruz in the eighth before Danys Baez recorded the final three outs for his first save.
Omar Daal (0-1) took the loss, yielding four runs and eight hits in 6 1/3 innings in his Orioles' debut.
The Indians took a 1-0 lead in the top of the second on Shane Spencer 's double and rookie Travis Hafner 's RBI single.
After the Orioles tied it in the bottom half, the Indians scratched for the go-ahead run in the third as Brandon Phillips led off with a single and Milton Bradley reached on an infield hit.
Vizquel dragged a bunt to the left of the mound and Phillips scored when Daal's throw sailed down the right field line. Vizquel was credited with a hit.
Celebrity Families of 2014 [PHOTOS]