KANSAS CITY — Dominic Smith’s speed may have turned the tide of the Red Sox’s playoff push.

The 224-pound first baseman isn’t known for his legs, or his ability to swim around the tag, but Smith displayed both to steal a crucial run in Boston’s 6-5 win over the Royals at Kauffman Stadium on Tuesday. The victory clinched the playoff tiebreaker over the Royals, with Boston taking four of five from Kansas City with one to play, and moved the Red Sox to a half-game behind Kansas City for the third and final AL Wild Card spot.

But without Smith racing down the third-base line, none of Boston’s movement in the standings may have taken place.

Smith stood at third with the score tied in the sixth when Royals starter Seth Lugo threw a curveball in the dirt, which got past Salvador Perez. Smith initially froze, unsure if the ball had rolled far enough away for him to break for it, but once he saw the ball destined for the backstop, Smith took off. Perez handled the ball and threw to Lugo in time to make a quick tag, but Smith turned his body, picked up his left arm and slid his right arm towards the plate.

The home-plate umpire called Smith out, keeping the game tied and leaving a runner on second with two outs, but Smith was called safe after review, giving the Red Sox a lead they would never relinquish.

“Dom wasn’t too sure, he thought he was out,” Red Sox manager Alex Cora said. “… You kept looking and looking at [the replay] and it was like, ‘Man, he got that in him — the swim move.’ Hesitation on the ball, you don’t want to make the out at the plate, but when you got to go, you go. … We needed every inch tonight.”

Smith added: “I didn’t really take off until I heard everybody screaming ‘Go!’ and then seeing the ball finally roll to the wall. Way too late. If you’re taking off when the coaches are saying go, nine times out of 10 you’re going to be out. So I guess it’s that one time when you’re not.”

The dugout erupted when the call was overturned, recognizing the significance of the moment in the crucial series. Smith though, was just relieved.

“I was just more thinking about like, ‘Damn, that’s a terrible read.’ It shouldn’t even have been a close play when the ball went all the way to the wall,” Smith said. “So that was my perspective on why I looked like that.”

But the swim move of the Red Sox’s season? Pure instinct, Smith said. He had watched the speedy runners such as Jarren Duran and David Hamilton do it on the basepaths, but it wasn’t something he had been practicing day in and day out.

“My slide is a little bit slower than other people’s, so I might have a little bit more time to move,” Smith joked. “So I think it’s just something where you’re trying to see his glove. You’re trying to touch the base before he touches you, and I was able to do that right there.”

Smith’s “freestyle” swim move as he called it was the encore to another speedy move on the basepaths. He also scored in the second inning by taking off on contact during Ceddanne Rafaela’s groundout to first base, scoring from third without a throw.

But it was the mad dash home in the sixth that helped seal it. After Smith scored, Rafaela walked, and pinch-hitter Rob Refsnyder was intentionally walked, setting up Masataka Yoshida for a two-run single to put Boston ahead 6-3. Yoshida had three RBIs on Tuesday after putting together a four-hit performance in the opener.

Kansas City rallied in the eighth to cut it to one, but veteran closer Kenley Jansen came in and shut the door with a four-out save, sealing Boston’s third straight series win with three strikeouts.

A playoff-type atmosphere has filled the air in the first two games at Kauffman Stadium because of the AL Wild Card race, and moments like Smith’s, and Jansen’s, are the types that help get a team into the postseason.

“These guys are getting the taste right now for it,” Jansen said. “Hopefully we keep grinding and stay hungry and take it one game at a time. It’s fun to be in this situation. I kind of love that you work the whole offseason for and build for this, and I’ve been in this situation many, many times. This is where it’s at. This is where I like.”