Desmond Jennings chipped in with two hits and four RBIs, and Ben Zobrist homered on a night Marlins' pitching combined to walk nine.
"I don't remember seeing a team struggle from the first to ninth hitter," Marlins manager Ozzie Guillen said. "It's kind of hard to figure it out. I'm not saying they're not trying. They come here every day early to hit. We give them the reports, everything. All of a sudden, they shut it down.
"I know we're facing a good pitching staff. But in the meanwhile, I don't think our at-bats are good enough to fight. If we want to compete, we need to hit. We need to get better offensively."
Miami's lone hit off was by Donovan Solano off Moore with one out in the first inning. It was a clean liner into center field, nothing to dispute.
From that point, Moore clamped down, striking out eight in seven innings.
In their history, the Marlins have been held to one or less hits 10 times. This was the eighth time they were limited to one hit, with the last being on June 28, 2011 at Oakland.
"[Moore] really settled in," Rays manager Joe Maddon said. "How about the curveball strike? That was a big difference in that once he settled in. I loved the fastball, but I liked the fact he commanded the breaking ball without trying to over-throw it, making it too good of a pitch.
"Then it started becoming a strike and now they have to honor fastball and breaking ball mentally. That makes it very difficult. So once he got in the groove with the breaking ball strike, it made it a little bit easier for him."
The Marlins, coming off a 1-8 homestand, have now dropped nine of 10. In doing so, their record fell to 32-32, marking the first time they've been at .500 since May 8 (15-15).
For Miami, the more pressing issue than the wild outing is Zambrano's health.
The right-hander exited his previous start, a June 9 loss to Tampa Bay, after 2 1/3 innings due to back stiffness. Zambrano gave up seven runs on five hits on that day at Marlins Park.
In his last two starts, the veteran has allowed 11 runs in 4 1/3 innings with nine walks.
Zambrano insists he is fine. The right-hander felt the ball was coming out of his hand well, and his velocity was there, in the low 90s.
"I walked too many guys today," Zambrano said. "If you have to blame somebody, blame this one on me. As a veteran on this team, I've got to be able to control the situation. I couldn't do that. The game is over. Unfortunately, we lost the game, but we also can't put our heads down. We have to keep our heads up and think that we have a very, very good ballclub. We have to come tomorrow and keep fighting."
Zobrist had a big night with a homer in the fifth off Chad Gaudin. He added a single and walk and scored two runs. In four games against the Marlins this year, he is 9-for-14 (.643) with three homers, seven runs and six RBIs.
Gaudin, although he gave up four runs, provided a big boost by working 4 2/3 innings, saving the bullpen.
The Rays pitchers, meanwhile, combined to strike out 12 batters. By fanning more than 10, it triggered a Tampa Bay promotion, where fans receive a free Papa Johns Pizza.
"We did two good things," Guillen said. "Gaudin pitched very well, and we filled them up with pizza. Thank the Miami Marlins. [The fans] are going to have pizza tonight. Hope they save me a little piece."
Zambrano labored through a 46-pitch second inning, walking four and allowing three runs.
The right-hander walked Zobrist and Carlos Pena to open the inning, and Jose Lobaton ripped a double down the right-field line. Lobaton has four doubles on the season, with three coming against Miami. He collected two at Marlins Park on June 8.
Trying to give his team some innings, Zambrano entered for the third inning, but he was lifted after two batters. Zobrist singled and Pena walked, prompting Guillen to go to Gaudin. The Rays made it 4-0 on Will Rhymes' two-out single, with the run charged to Zambrano.
In their last 10 games, the Marlins have been outscored 67-17.
"My sinker was running too much today, but it's not an excuse," Zambrano said. "It was a bad game. I take the blame."