LOS ANGELES -- Clayton Kershaw may have a long ways to go to become the best pitcher in Dodgers history, but on Monday night he had the thrill of celebrating his team's National League Division Series victory alongside the current holder of that title.

Moments after the Dodgers' 4-3 National League Division Series-clinching victory over the Braves, Kershaw -- who did not allow an earned run over six innings of work on just three days' rest -- was greeted in the Dodgers' clubhouse by an embrace from four-time World Series champion Sandy Koufax.

"He's just happy for us. Sandy is such a great guy. I think he genuinely cares about, not only this team, but kind of our well being," Kershaw said. "He cares about us. That's awesome. I have a great relationship with him. I have a ton of respect for him.

"To get a hug and get a, 'Good job,' from a guy like that, from a guy that's been there, from a guy that's done this before and was the best at it for a long time, is pretty special."

At just 25 years old, Kershaw has already had some pretty special moments in his career, as well.

As the NL Cy Young Award favorite this season, Kershaw could be quickly closing in on Koufax's franchise record of winning three Cy Young Awards. Kershaw previously won the honor in 2011 and finished second in the voting a season ago.

He added another accomplishment to his resume on Monday, pitching on short rest for the first time in his career -- and pitching well, at that. Though he ultimately settled for a no-decision, Kershaw limited the Braves to just two runs, both unearned, off three hits while striking out six over six innings of work.

"Three days' rest, 30 days' rest, he was the same guy," catcher A.J. Ellis said. "Same guy he's been the entire season. This guy's a warrior. He's the best pitcher on the planet. There's nobody on the mound that I'd rather catch."

And apparently there's nobody the Dodgers would rather have on the mound.

With Los Angeles holding a 2-1 series lead heading into Monday's Game 4, manager Don Mattingly turned to his ace on short rest in an attempt to close out the series and avoid a trip back to Atlanta. Despite Kershaw never having pitched on only three days' rest, Mattingly said he fully expected to get this type of performance out of the dominant southpaw.

"Clayton did an unbelievable job. For a guy pitching on three days' rest like that, and he basically didn't give up anything," Mattingly said. "He was just tremendous. It told us that we made a good decision there with him. He did his job."

If Kershaw had his way, he would have tried to do a lot more. As is his nature, Kershaw said that he would have liked to stay in the game beyond his six innings and 91 pitches, but Mattingly did not want to stretch him out too much given that Kershaw had thrown 124 pitches in Game 1.

"He wasn't hurting or tired or anything out there," Mattingly said. "So he would have kept going, he could have kept going and he wanted to keep going -- we just wouldn't let him."

Kershaw admitted that he may have taken pitching in the postseason for granted with the Dodgers reaching the playoffs in each of his first two seasons in 2008-09. After three straight seasons of not qualifying for the playoffs, however, he's prepared to never let that happen again.

Kershaw has been vocal about wanting the ball anytime Mattingly calls his name and not letting go of it until the skipper all but pries it out of his hands.

"My first two years I kind of just thought that always happened, I guess. I'd always get to go [to the postseason]," Kershaw said. "The last three years of not getting to go really taught me to embrace this opportunity. It kind of goes into what I'm saying all along. You never know when this is going to happen again. You've just got to enjoy it, savor it."

So that's exactly what Kershaw did on Monday night, sharing the moment with Koufax, as well as his Dodgers teammates, coaching staff and front office personnel.

As for what's next, Kershaw said he is fully prepared to take the hill on short rest again if the situation arises in the National League Championship Series. Game 1 of the NLCS isn't until Friday and the Dodgers will likely turn to co-ace Zack Greinke for the series opener, but Kershaw will be ready to go long before that series even begins.

"If Donnie wanted me to pitch tomorrow, I would do it," Kershaw said. "This is the postseason. All that other preparation, 'Is he going to be ready, is your arm going to be tired?' -- throw that out the window. It doesn't matter. You just go. It's a one-month sprint and I'm looking forward to the next couple games."