We remember Keith Foulke gloving that one-hopper back to the mound in 2004 and oh-so-carefully flipping the ball to Doug Mientkiewicz at first base before he turned to hug Jason Varitek.

In 2007, Jonathan Papelbon was on the mound for the Red Sox and ended the game with a strikeout. He flung his glove in the air, tore off his cap, and grabbed Varitek with both arms.

The lasting image in 2013 is that of Koji Uehara pointing to the sky as he leapt into the arms of David Ross after a strikeout.

For a closer, the ultimate glory is getting the final out of the World Series. It’s the video you never tire of, the photograph that hangs over the fireplace. It’s a legacy nobody can question.

For Craig Kimbrel, it’s the dream he’s had since childhood.

“You want the ball in the last inning to get the last out,’’ he said. “That’s the goal for any relief pitcher. I think about that all the time. Nothing I could ever accomplish in baseball would be better than that.’’

As the Red Sox enter the postseason Thursday with Game 1 of their Division Series against Cleveland, Kimbrel could well author the same kind of moment Foulke, Papelbon, and Uehara did.

His 256 saves since 2010 are the most in the majors and the most in a player’s first seven seasons. Kimbrel has been an All-Star in five of his six full seasons and has averaged 14.5 strikeouts per nine innings.

But Kimbrel also enters the postseason as a player the Red Sox can’t be certain of.

In his last three appearances, the righthander has faced 13 batters and retired only six of them. He walked six and allowed five runs to score.

On Saturday, after Kimbrel gave up a run in the ninth inning of a 3-3 game against Toronto, manager John Farrell was asked if he could consider demoting his closer and putting Uehara back in the job.

“Not at this moment,’’ Farrell said.

That Farrell did not dismiss the notion was telling. A closer with control problems is postseason poison, and Farrell could well turn away from Kimbrel if he struggles against the Indians.

During the regular season, managers are careful not to bruise egos. In the playoffs, decisions drop like sledgehammers. Farrell replaced third baseman Will Middlebrooks with rookie Xander Bogaerts in the 2013 ALCS, then in the World Series benched catcher Jarrod Saltalamacchia after Game 3 in favor of Ross.

Kimbrel may not require such drastic action. Pitching coach Carl Willis said Kimbrel needs an adjustment of his posture. In simple terms, he’s moving side to side on the mound instead of staying behind the ball. That has affected his ability to throw strikes.

“It’s something he can correct very quickly,’’ Willis said.

There’s proof of that, too. Kimbrel walked four batters on Aug. 9, then walked four in the 17 appearances and 15 innings that followed. He allowed one earned run in those games.

Farrell also won’t be quick to forget that opponents have hit .152 against Kimbrel this season; or that his pitches are equally effective against righthanders and lefthanders.

“This is a special guy,’’ Farrell said.

Kimbrel averaged 97.3 m.p.h. with his fastball this season and threw his curveball at 86.6, extraordinarily hard for a breaking ball.

Among relievers, Kimbrel was fourth in average fastball velocity. None were close to him in terms of curveball velocity.

“That curveball is what sets him apart,’’ said teammate Aaron Hill, who faced Kimbrel 10 times in previous seasons. “Everybody throws hard these days. But he pitches because of that curveball; he’s not just a thrower. Not too many closers can do that. He’s an animal.’’

Farrell contends he can hear a buzzing sound when Kimbrel throws his curveball particularly hard.

“It’s kind a freaky,’’ he said.

Veteran catchers Ryan Hanigan and Bryan Holaday said that’s an exaggeration.

“It’s a great pitch,’’ said Hanigan, “but there’s no sound. If a pitch made a sound, I would know. That hasn’t happened yet.’’

Said Holaday: “If he’s throwing his curveball for strikes, it’s game over. You have to respect 98. If you do look for a curveball, you still can’t hit it. He doesn’t give a hitter much to work with.’’

Kimbrel developed his curveball in 2008 while pitching for Wallace State Community College in Alabama.

“It progressed as the years went by,’’ he said. “It’s been a slider more than a curveball at times. It almost depends on the day. If I can throw that for a strike early in the count, I have the advantage.’’

Don’t question his determination to get right in a hurry. Kimbrel had surgery on his left knee July 11 to repair damaged cartilage. He was back on the mound July 30 in a minor league game and came off the disabled list Aug. 1.

Kimbrel started throwing from his knees three days after surgery to keep his arm strength intact. He has converted every save chance since coming back.

“It was frustrating to sit back and watch,’’ he said. “I don’t normally do that. But I enjoyed watching us play and being able to really appreciate the individual guys on the team.’’

Kimbrel, who is signed through 2017 with a team option for 2018, hopes to stay with the Sox long-term. At 28, he should be able to maintain his elite velocity for another few years.

He also likes his new city.

“There are a lot of perks here,’’ he said. “I take my bicycle for rides along the Charles. I’ve biked around a lot of the city. It’s beautiful.’’

Kimbrel and his wife, Ashley, enjoy eating out in the North End, although he lets her wait in the long lines at Mike’s Pastry on Hanover Street.

“I can’t believe that place,’’ he said. “Everybody goes there.’’

If Kimbrel regains his control and is able to get that final out of the World Series a few weeks from now, he may never wait in line in Boston again.

“That’s really all I want,’’ he said. “I told Koji in spring training that I saw him do it and I want my turn. That would be the ultimate.’’

Peter Abraham can be reached at pabraham@globe.com.