Moore (4-0) turned in his fourth straight dominating start to begin the season, allowing one run on just two hits. The left-hander walked three and struck out nine while lowering his ERA to 1.04.
Roberts went 3-for-4 with solo shots in the first and third innings while Yunel Escobar added a two-run blast for the Rays, who picked up their fourth straight victory in taking the opener of a three-game series.
Sean Rodriguez contributed an RBI triple during Tampa Bay's four-run first.
Yankees starter CC Sabathia (3-2) was touched for five runs on seven hits in a seven-inning effort.
Robinson Cano collected both New York hits in the loss, including a bases-empty homer.