An MRI confirmed the sprain. Kluber is expected to miss four to six weeks.
"It's disappointing in a lot of ways," Francona said following his club's 5-1 loss on Tuesday. "We're right in the thick of things, and we're running Corey Kluber out there, feeling pretty good about ourselves."
At 62-51, the Indians are five games behind the Tigers in the American League Central and one game out of an AL Wild Card spot. Kluber has played a key role in keeping Cleveland in contention.
Kluber is 7-5 with a 3.54 ERA through 21 games, 19 of them starts. Across 122 innings -- the second most on the team, behind Justin Masterson -- Kluber has 116 strikeouts against 26 walks, and he has kept hitters to a .258 batting average.
The Indians won five of Kluber's last seven starts. In his last six turns, he had a 1.98 ERA and held batters to a .213 average. Kluber had 40 strikeouts and 10 walks over the 41 innings of those outings. He was 5-3 with a 3.07 ERA in his last 16 appearances.
The news of Kluber's injury compounded the current woes of the Indians, who began this week's crucial series against the division-leading Tigers with a pair of losses.
"It just kind of seems like after yesterday and today, it's just been one bad thing after another," Nick Swisher said on Tuesday. "He's been such a huge part of this rotation. He's been such a huge part of this team.
"To lose a guy like that, especially running down the stretch, that's the last thing you want to hear. ... We just hope for a speedy recovery and to get him back as soon as we can."
Fellow starter Zach McAllister sat out seven weeks with his own sprained right middle finger earlier this season. The Indians were cautious with McAllister, who had a sprained pulley in the finger, and intend to treat Kluber with similar care.
As did Kluber, McAllister felt pain in his finger after throwing breaking balls.
"Our hope is that it won't be as bad as McAllister was," Masterson said Tuesday. "He was out for quite some time. Hopefully, it can be one of those where we can take a few weeks, get him back on track and have him, because he'll be a big part as we continue on."
A plug for the hole in the rotation that Kluber's injury created might come in the form of right-hander Danny Salazar, the organization's No. 6-ranked prospect, who was to be called up from Triple-A Columbus on Wednesday to make his second Major League start.
In his first outing, Salazar turned heads by holding Toronto to one run and two hits over six innings. Cleveland has monitored Salazar's innings all season, as he underwent Tommy John surgery in August 2010.
Francona did not say if Salazar would stick in the rotation after Wednesday, but he did reveal that Scott Kazmir will take Kluber's scheduled turn on Saturday.
"I think our player-development people have done a really good job managing [Salazar's] innings," Francona said. "We'll bump Kaz up. Kaz lost his couple days' rest, and he's OK with that."
Langwell would seem like a logical choice to be sent out on Wednesday to make room for Salazar. He is 1-0 with a 5.06 ERA in 5 1/3 innings this season; he did not pitch on Tuesday.
"I'm disappointed for [Kluber]," Francona said. "It makes it harder for us. It doesn't mean that we can't do it, it just makes it harder. But I'm real disappointed for him, because he's worked so hard. And he will continue to, and he'll be fine. Just, you've got to be patient with these types of things or you can get yourself in trouble, so we won't."