"The hits were coming so fast, I didn't have time to think," Zobrist said. "Hitting was so contagious at that point, he threw a pitch and I swung."
Perez (0-2) came in seeking his 25th straight save since blowing a late lead on Opening Day to Toronto.
Then Zobrist delivered.
"That was some win," Rays manager Joe Maddon said. "It was Rhymes' first home run since 1837, and Carlos has hit probably two balls that way all year."
It was Rhymes' second career homer. The first came in 2010.
"I knew I got it," he said. "I don't hit many, but when I do, I can feel it."
"I started getting loose when Rhymes hit the homer," Rodney said. "I told the guys, 'Don't worry, we're going to win this game.'
"This is a big win for a young team, perfect to get us going again."
The Rays earned a four-game series split with their fourth win in 14 games.
"It's been good so far but obviously, the first day and the last day of the first half stunk," Perez said. "I made some bad pitches and they hit them."
Rodney had problems, too. He got two quick outs in the bottom half before Brantley singled and went to third on a single by Carlos Santana. The right-hander then got Kotchman to bounce into a game-ending forceout at second.
"I just kept throwing strikes, that's all you can do," Rodney said. "No worries."
Kotchman had three RBIs against his former team as Cleveland took a 5-0 lead against starter James Shields.
Indians starter Zach McAllister didn't allow a hit until No. 9 hitter Rhymes' one-out single in the fifth. Luke Scott and Desmond Jennings each had two-run doubles in the sixth, when the Rays scored four unearned runs.
Kotchman had an RBI single in the second, opened the fourth with his eighth homer, and singled home another run in the fifth. The first baseman hit .306 for Tampa Bay a year ago and signed with Cleveland as a free agent in February.
Johnny Damon, the former Ray who signed with the Indians this spring, tripled to open the third. He scored when Shields threw the ball away on a pickoff attempt. Damon trotted home on the error for a 2-0 lead.
Kipnis singled to start the fifth and later scored on Kotchman's third hit for a 5-0 lead.
All-Star shortstop Asdrubal Cabrera's error opened the door for the Rays in the sixth.
Pena got the Rays' second hit, beating out a slow roller to second with one out. Zobrist followed with a grounder to second baseman Kipnis, whose throw to second was dropped by Cabrera. With two outs, Scott doubled off the right-center wall for two runs.
After McAllister walked Keppinger, Jennings' two-run double to the right-center gap made it 5-4.
Shields gave up 10 hits and four earned runs over seven innings, throwing 121 pitches. It was the third game in a row and fifth time overall that Shields allowed 10 or more hits.
Maddon marveled at Scott, who missed three weeks with a sore back and endured a team-record 0-for-41 slump.
"Through all this, the guy has 42 RBIs," Maddon said of Scott's team-high total.
Maddon said the Rays should not worry about when three-time All-Star Evan Longoria may return from a hamstring injury and believe they are good enough to overcome injuries. Tampa Bay sent 14 players to the disabled list in the first half.
"We have to play better and we can," Maddon said. "We can hit anybody, anywhere, anytime. It is not time to tread water. We have to swim like we're in the Olympics and get after it."
The Rays' two errors gave them 71, second-most in the majors to Baltimore. ... Cleveland put LHP Nick Hagadone on its minor-league disqualified list, one day after optioning him to Triple-A Columbus. General manager Chris Antonetti said Hagadone suffered "a self-inflicted injury to his pitching hand" after leaving a game Friday. He won't get paid while on the disqualified list and may require season-ending surgery. ... Jennings, batting .161 in his last 20 games in the leadoff spot, was moved to seventh. ... Maddon said RHP Jeremy Hellickson will pitch the first game after the All-Star break Friday against Boston. All-Star LHP David Price, expected to work in Tuesday's Midsummer Classic, will start Saturday.