It was the sixth win of the season for Moore, but just the second in his last five starts.
The Rays' bullpen was a concern coming into the matchup after five relievers were needed in Monday's game. Rays manager Joe Maddon used four relievers on Tuesday, and they combineded to finish the last four innings without allowing a run.
"That was our whole win," Moore said about the bullpen. "They definitely saved our [rear ends] as far as coming in and covering four innings against a squad like that. They're playing well and they are a pretty good disciplined hitting team, so hats off to them and the countless times they've picked me up this year."
Batting leadoff, B.J. Upton went 2-for-4 on the day with a run.
"We've kinda done it, put it together here and there," Upton said about responding after losing the last two games. "We just haven't really had our big streak yet, and it's gonna happen. If we keep doing the little things like we have been lately, we'll be playing some close ballgames."
Outfielder Matt Joyce, who was activated from the disabled list prior to the game, went 0-for-4 at the dish with two strikeouts.
Tampa Bay didn't take long to get on the scoreboard. Carlos Pena knocked his 14th homer of the year over the right-field fence after a single from Upton in the first inning. The long ball was the sixth in Pena's last 25 games.
"The one to Upton was a 2-0 fastball just right down the middle of the plate," Tomlin said. "The pitch to Pena was, I think maybe away, but it was up, and he did a good job of getting extended and hit a home run."
Ben Zobrist followed that with a walk before Tomlin recorded a pair of outs. But Luke Scott was able to pick up his team-best 43rd RBI of the season when he roped a line drive to right field that Shin-Soo Choo had problems fielding, resulting in a triple and 3-0 lead for the Rays.
"You try to do your best to limit the damage as much as you can," Tomlin said. "That's kind of part of the game sometimes. Mistakes beat you and that's unfortunately what happened for us today."
Cleveland got its first run in the top of the fourth when Jose Lopez hit a sacrifice fly to deep right field with the bases loaded and no outs that brought in Asdrubal Cabrera, who singled to lead off the inning. The Tribe couldn't capitalize further as Moore struck out Carlos Santana before forcing Shelley Duncan to fly out to center.
The Indians added another run in the top of the fifth when Jason Kipnis singled home Lou Marson with two outs. Moore then walked the bases loaded, but was able to get out of the jam when Lopez flew out deep to left field.
"He almost crushed that ball, but he did pitch through it, and those are the kind of growth moments we're always looking for with our younger guys," Maddon said. "When you're faced with that kind of competition, I don't think you ever become fortunate, you have to really work through it, and I thought he held his own pretty well right there."
The Rays pushed the lead back to two in the bottom of the sixth with two outs. After a fielder's choice put him on first, Desmond Jennings stole second, and a single by Jose Lobaton was enough to bring him in and give Tampa Bay an insurance run.
"He's really fast," Maddon said, regarding Jennings' stolen base. "He's very instinctive and a good baserunner on top of that. I know he's fast, but he runs the bases well. ... He has that component to his game."
Cleveland then turned to its bullpen for the rest of the night, as Tomlin left the game after 5 2/3 innings of four-run baseball on seven hits and a walk.