Moore (1-0) allowed the two Cleveland hits over six innings while striking out eight. Jake McGee, Brandon Gomes and Joel Peralta worked an inning each to complete the shutout.
Zobrist delivered a run-scoring double in the fourth inning and came home later in the frame on a ground out by Yunel Escobar. Another double by Zobrist brought in two runs in the fifth. He has seven RBI thus far this season and is batting .467.
Zach McAllister (0-1) took the loss, allowing all four runs on six hits in six innings.
Both teams have split their first four games of the season. Cleveland is starting its second straight series on the road to open the campaign while the Rays have have yet to leave home.
Celebrity Families of 2014 [PHOTOS]