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Moore (1-0) limited the Indians to a pair of hits by Michael Bourn, walked two and struck out eight. He retired his final eight batters and came out after 100 pitches.
"It was the perfect time to get him out -- 100 pitches, six innings," manager Joe Maddon said. "He goes home feeling really good about himself. That's a great game for him to build on."
Moore has been a notoriously slow starter throughout his pro career. He was winless in four April starts in his first full season in the majors. Including five minor league seasons, he entered Friday with a 4.52 ERA in April and May compared to 2.30 ERA over his career in June to October.
"It feels good for us to have a combined shutout like that. It's something we really work towards every game, and to have the first one of the year on my night feels nice," Moore said.
The lefty struggled with command of his fastball for most of spring training, but began to feel better about his performance in his final exhibition start last weekend.
"It's really just a matter of keeping that tempo where it was in that exhibition game against Detroit. I felt like it was right there," Moore said. "Just by the results of that outing, it felt like something picked up. For me, it was my tempo. That cleaned up everything."
Zobrist, who has driven in a team-leading seven runs in four games, did most of the damage against Zach McAllister (0-1) with a pair of doubles.
Bourn reached on an infield single in the first and doubled into the left-field corner in the third. Only two other runners reached base against Moore, who walked Mike Aviles in the third and Michael Brantley in the fourth.
"He wasn't throwing anything up there that we haven't seen. I think we didn't take advantage of, when we had guys on base and didn't come up with the big hit," Cleveland's Jason Kipnis said.
"He had his off-speed stuff working. Pounding the zone. Working ahead of guys," the Indians' Nick Swisher added. "He really, really kind of got into that groove, and made it a tough day."
The Indians, who were 0 for 5 with runners in scoring position, struck out 11 times and didn't have a baserunner after the fourth inning.
Two of Tampa Bay's four runs off McAllister were unearned. Cleveland's starter allowed six hits, walked none and struck out three over six innings.
Sam Fuld led off the Tampa Bay fourth with a single and scored on Zobrist's double down the right-field line. Yunel Escobar drove in Zobrist from third with a grounder to short to make it 2-0.
A two-out fielding error by Aviles at third opened the door for the Rays to add two more runs on Zobrist's fifth-inning double.
Zobrist has hit safely in all four of Tampa Bay's games. He had a solo homer and sacrifice fly on Opening Day, then added a pair of RBI singles the following night.
"He went to the WBC, and I think that's a real important part of his start right now," Maddon said. "He really takes that kind of thing seriously, so he came to camp ready to play."
- The Rays put RHP Jeff Niemann on the 15-day disabled list because of soreness in his pitching shoulder. The move was retroactive to March 28. Niemann, moved to the bullpen after competing with Roberto Hernandez for the final spot in the starting rotation this spring, has yet to appear in a game this season. Gomes was recalled from Triple-A Durham to fill the roster opening.
- Cleveland manager Terry Francona said RHP Carlos Carrasco, currently serving a six-game suspension dating from July 2011, will start Tuesday against the New York Yankees. Carrasco missed the end of 2011 and all of 2012 after elbow surgery.
- Indians 1B-DH Jason Giambi, on the 15-day DL with a lower back strain, is working out with the team and could be activated Tuesday.
- Cleveland LHP Scott Kazmir, a one-time Tampa Bay ace who was scratched from his scheduled start Saturday because of a strained ribcage muscle, didn't make the trip. He is undergoing treatment in Cleveland.
- The Rays traded minor league C Stephen Vogt to Oakland for a player to be named or cash.
- The Cleveland Plain Dealer reported negotiations on a multiyear contract between the Indians and Kipnis ended when the regular season began.