Ramirez connected for a solo shot in the first inning off Matt Dermody (0-1), belted a two-run shot -- his 200th career homer -- in the third and added another solo blast in the sixth inning, when the Guardians hit for the cycle, had eight straight hits and scored five times.
Ramirez came up a second time in the sixth bidding for a rare four-homer game but struck out swinging. The four-time All-Star third baseman had another chance in the eighth but grounded to third.
"My last two at-bats I was swinging away," Ramírez admitted.
It was Ramirez's first career three-homer game, and it was the seventh time in his career that he has hit a home run from both sides of the plate, according to ESPN Stats & Information.
"Everyone in [the clubhouse] would say the same thing, he's the best player in the game," said Cleveland starter Aaron Civale (2-1), who had his second straight solid start. "He plays the game the best way every single night.
"He might not be on paper the most talented. He might not be considered the best. But when he plays the game the way he plays it, he's the best in the game."
Ramirez also showed off his defensive prowess in the sixth by backhanding Justin Turner's hard shot and throwing off-balance across the diamond to get Boston's third baseman while backpedaling in foul territory.
"If that was back in the day like Brooks Robinson, that play would be shown over and over and over," Guardians manager Terry Francona said.
Will Brennan added a homer as the Guardians won the series and handed the Red Sox their fifth loss in six games. Boston is under .500 for the first time since April 28.
"It's not the type of baseball we want to play," manager Alex Cora said. "But here we are."
Triston Casas chased Civale in the sixth with his seventh homer to pull Boston within 3-2.
But Cleveland, whose offense has awakened after a slow start, pounded former teammate Corey Kluber in the sixth. After Ramírez homered, Andres Gimenez hit a two-run double, Myles Straw hit an RBI triple and Brennan singled home a run as the Guardians opened an 8-2 lead.
Kluber, who was recently demoted to the bullpen, was tagged for seven runs and 11 hits in 3⅓ innings. When Cora pulled him, Kluber, a two-time Cy Young winner with Cleveland, made a slow walk from the mound where he had so much success to Boston's dugout.
"It's really tough," Cora said about seeing Kluber hit so hard. "He couldn't buy a break."
The Associated Press contributed to this report.