But Mujica's pitch was badly mislocated, and Asdrubal Cabrera nullified manager John Farrell's defensive alignment with a three-run walk-off homer to right.
A 7-4 win by the Indians took 12 innings -- this after a rain delay of two hours and 28 minutes before the game could start.
"It was a changeup, high in the zone," said Mujica. "That situation right there, I'm looking for a ground ball. I want to throw my best pitch. The first one was pretty good, and [the second] one just ran a little bit inside, and he put a pretty good swing on it."
For the Red Sox, it was the end of a thoroughly frustrating three days in Cleveland that ended in a sweep.
The loss took 4 hours and 29 minutes, and the Red Sox, who came to town with a seven-game winning streak, will now try to regroup during Thursday's off-day in Detroit in advance of a weekend showdown with the Tigers.
"Regardless of the time of the day, any time you go 12 innings, they sting a little bit," said Farrell. "We've got a day to recover and regroup from a bullpen standpoint, so that's the sole reason [Koji Ueahara's] out there for two innings tonight."
Yes, Uehara was asked to record more than three outs for the Red Sox for the first time since April 26.
Burke Badenhop, Craig Breslow and Junichi Tazawa also were strong out of the bullpen for Boston.
Mujica's night started with a strikeout of Ryan Raburn. But Mike Aviles spun a grounder deep in the hole at shortstop that wound up a single.
Up stepped Michael Bourn, and he drilled a grounder by a diving Brock Holt, who was playing just the fourth game of his career at first. That single put runners at the corners with nobody out.
"If you're over there, coming off the bag, you've got to stay low," said Holt. "He hit it pretty hard. The grass was pretty wet, so it got on me pretty quick. It skipped over my glove. I was there far enough, but it just slipped over my glove. That's the first time I've had to get a ball like that. I dove, tried to keep it in, but it just skipped past me."
Down, 4-2, after six innings, the Sox came back to tie it in the top of the seventh on a two-run single by Holt.
Stephen Drew started the rally, working a walk after falling behind 0-2 against tough Indians righty Corey Kluber. Daniel Nava followed with a single, his first hit for the Red Sox since April 21.
After a sacrifice bunt by Jackie Bradley Jr., Holt got the job done, knocking Kluber out of the game.
"Jackie lays down a good bunt to advance a couple of runners, and Brock with a bit base hit up the middle for two runs and a tie score," said Farrell.
Kluber stifled Boston early, but David Ortiz came through in the sixth, drilling a two-run homer to right that gave Boston a 2-1 lead.
"David gets an 0-2 pitch that he runs out of the ballpark to give us a boost of energy, and obviously a short-lived lead, but a lead nonetheless," said Farrell.
Prior to that at-bat, Ortiz had been in a 9-for-62 funk since May 15, with one homer.
The Indians rallied right back in their half of the sixth. Starting pitcher Brandon Workman walked Cabrera, after which catcher A.J. Pierzynski was ejected for arguing with home-plate umpire Quinn Wolcott.
Michael Brantely followed with a single, and that was all for Workman.
Lefty Chris Capuano came on and did not get the job done. Jason Kipnis geeted him with an RBI single. Lonnie Chisenhall reached on a bunt single to load the bases. David Murphy lined a single out of the reach of Dustin Pedroia and into right for a two-run single to give the Indians a 4-2 lead.
Badenhop prevented the inning from getting worse, inducing Jason Giambi into a 4-6-3 double play to end the inning.
Workman went five-plus innings in his third start, allowing four hits and three runs. He walked two and struck out three.
"I felt pretty good. Obviously the leadoff walk in the sixth wasn't the way I wanted to start that inning, but overall, I felt pretty good," said Workman.
The Indians struck for a quick run in the bottom of the first after Cabrera reached on a one-out single that Bradley and Drew converged on.
Cabrera stole second and scored on an RBI single by Brantley.
Ultimately, it came down to Cabrera again -- several hours later.
"I was just trying to hit a fly ball to the outfield," said Cabrera. "I know Mujica. I know what kind of pitcher he is and what he likes to throw. He threw what I was looking for, and I hit it pretty good."