Matt Kuchar hits his tee shot on the 17th hole during the final round. (Andy Lyons/Getty Images)

It would be hard to discern whether Matt Kuchar, whose lips rarely seem to cover his teeth, had just made birdie or bogey, had just enjoyed a fine steak with his best friends or blown out his tire and was left stranded by the side of the road.

“He’s always smiling,” said Zach Johnson, a neighbor and friend from St. Simons Island, Ga. “I think that’s the beauty of Matt.”

Sunday evening, as he stepped to the 18th tee at TPC Sawgrass’s Stadium Course, he could grin even wider and ignore the water that threatened all down the left-hand side of the fairway. He needed only to make bogey to win the Players Championship, a cushion provided both by his own steely birdie putt two holes earlier and missed opportunities by his competitors. And when he closed out the victory by tapping in, the pearly whites came out in full glory even before his two sons could leap into his arms, followed by his wife — their mother — on Mother’s Day.

“It is completely a natural reaction,” Kuchar said, grinning again. “I love playing the game of golf.”

He loved it as much as ever Sunday. The win was, by a wide margin, the most significant of Kuchar’s career. At 33, he is old enough to have been a sensation as an amateur paired with Tiger Woods at the 1998 Masters; battered enough to have lost his playing privileges at the highest level and been relegated to the minor league Nationwide Tour; resilient enough to have fought his way back and become a top-10 machine, even winning the money title in 2010. But for all the times Kuchar’s smile has appeared on televisions as a man in contention — including the Masters a month ago, in which he was briefly tied for the lead on the back nine and finished two shots out of a playoff — he really hadn’t approached this sort of accomplishment.

So there was a contradiction: Was Kuchar so happy — wealthy, with a wonderful family — that he didn’t truly care to achieve?

“He is as gritty a competitor, and as fierce a competitor, as I’ve ever been associated with,” said Johnson, who tied for second. “His game, it models consistency. It models, I’d say, tenacity when it comes down to finishing the deal.”

The particulars of this victory: Kuchar closed with a 70 to finish at 13-under-par 275, two shots better than Johnson, Ben Curtis, Martin Laird and Rickie Fowler, who was seeking his second consecutive victory on the PGA Tour. And as much as Kuchar’s 15-foot birdie putt on the par-5 16th — a roll that gave him a three-shot lead — defined the victory, what transpired in the minutes and hours before at 18 made that possible.

Start with Laird, the Scotsman who Kuchar beat for his most recent victory, the Barclays in 2010. He missed the green and left himself eight feet for par to get in at 12-under 276. He missed, and closed with bogey.

“Just misread it,” Laird said.

Fowler, the 23-year-old fan favorite, came to 18 after making birdie at the island green 17th, and needed one more to get to 12 under. He laced a stellar approach to nine feet.

“Just pushed the putt a little bit,” Fowler said, and a pressure-applying birdie became a mundane par.

Over the final hour, those two putts loomed as the only chances to put real heat on Kuchar. Kevin Na, whose utter inability to pull back his club in a timely manner became the talk of his sport over the weekend, succumbed to the gremlins in his head Sunday, and merely became a spectator to Kuchar’s ride. After a steady start, the 28-year-old who led after 54 holes bogeyed four of the final five on the front side to plummet from the lead. By the time he dunked his tee shot at 13, he had heard more than his share of displeasure from the galleries, a rarity in golf.

“When I’m over the ball, it’d be nice if it was quiet,” said Na, who acknowledged time and again that he needs to improve his pre-shot routine. “Just guys, you can hear them talking. ‘Pull the trigger. Pull the trigger. Hit it.’ ”

Though he shot 76 and finished tied for seventh, five strokes back, Na left an imprint on the tournament. Slow play, an issue on tour as far back as anyone can remember, is now back under intense scrutiny. Woods, who shot an indifferent 73 to finish tied for 40th, said the problem is “worse” now than it was four years ago. His solution: “Very simple. If you get a warning, you get a penalty [shot]. I think that would speed it up.”

The last penalty shot on the PGA Tour was issued in 1992. Though Fowler and his playing partner, Curtis, were already done with their round before Kuchar and Na stepped to the 18th tee, there wasn’t one issued Sunday.

When Kuchar got to that final tee box, he could forget his three-putt bogey at No. 17. Because Laird had missed his par putt at 18, because Fowler had missed his for birdie, Kuchar could play for bogey, and know he would still win. So he smiled.

“Not a whole lot gets under my skin,” he said. With the biggest win of his career in the bag, why would it?