Young children, especially, can be brutally honest, in a totally innocent way. Or, in Fredrik Jacobson’s case, they can serve as the ultimate motivation.

Watching a televised highlight show at the end of last year’s PGA Tour season, with scenes of each winner raising a champion’s trophy, Jacobson was asked a simple question by his then-4-year-old daughter, Emmie: Why don’t you have one?

It was there, in their Hobe Sound, Fla., home, that Jacobson turned to his middle child.

“I told her, ‘I promise you I’ll get one this year for you.’ And it’s been haunting me,’’ Jacobson said. “I’ve been asked so many times by the kids, ‘Did you get a trophy this week, Daddy? Did you get a trophy?’ ’’

It was well past 1 a.m. in his native Sweden when Jacobson finally got to make the emotional phone call he’s been waiting for, unsure if his children were awake or aware of what took place a continent away.

Daddy’s got your trophy, Emmie. Sweet dreams, indeed.

Jacobson’s family returns to Sweden every summer once school gets out, and he’ll join them after this week’s PGA Tour event in Pennsylvania. He’ll be able to personally deliver the promised hardware impressively earned yesterday, when he captured the Travelers Championship by one stroke for his first PGA Tour victory.

It wasn’t easy. Jacobson shot a 4-under-par 66 at TPC River Highlands to finish 72 holes at a robust 20 under par. But he needed an 18th-hole bogey by Ryan Moore — the lone blemish in a superb 63 — to break a tie and put Jacobson in front. Handed the lead while he was walking up the 18th fairway, Jacobson knew what to do from there: a safe approach to 14 feet, a stress-free two-putt. It culminated a journey that began with overseas success — he made the move to the PGA Tour for the 2004 season, after winning three times the year before on the European Tour. Trips from Europe, though, can take a while. This one was no different.

“I didn’t know it was going to take this long before I won one,’’ Jacobson said. “But all the sweeter when it comes together.’’

There have been close calls. Jacobson has three runner-up finishes in his tour career, and had been solid this year: a tie for ninth at Bay Hill, a tie for fifth at the Texas Open. He was second for part of the final round recently at the US Open, roughly 2 miles behind Rory McIlroy.

Playing a course that once again wasn’t offering much resistance, Jacobson did his part, making five final-round birdies and never losing the lead.

But others were charging hard. Moore began four shots back, but caught Jacobson with a 10-foot birdie on the par-3 16th. Earlier, Michael Thompson tied Jacobson with a birdie on the 18th hole, his final stroke in a 62 that left the PGA Tour rookie alone in fourth, two behind.

John Rollins birdied the 18th for his own 63 to reach 19 under, and with Moore bunkered behind him in the fairway and Jacobson faced with a 7-foot putt for par, it appeared there might soon be a three-way tie for the lead.

Jacobson, though, holed his par putt to stay at 20 under. Moore went from the fairway bunker to a greenside trap, then blasted out to 7 feet. Needing to hole it to remain tied, Moore pushed it slightly, his ball catching a sliver of the right lip. Bogey.

“That 18th hole is going to sting a little bit, but any time you shoot 63 in the final round, not a whole lot to complain about,’’ said Moore, who went 64-63 on the weekend and now has three top-four finishes in six Travelers appearances.

About the only thing Jacobson didn’t accomplish was finishing 72 holes without a bogey. The only one he made came on No. 10 yesterday, when he slung a mud ball from the fairway way right of the green and failed to save his par. That opened the door for Moore (back-nine 32), Rollins (30), and Thompson (29) to make a late run, but it was the Swede smiling at the end, first to the finish line.

“I’m just enjoying this moment now,’’ Jacobson said. “I have no idea what tomorrow brings, but this is something I’m enjoying dearly. A lot of work and a lot of patience has come together.’’

And to the victor go the spoils. The $1.08 million is nice, as is the likely berth in next month’s British Open.

Those are secondary, though. Now that a promise has been kept, Emmie will be thrilled with her present. But Alice is 7 and Max is 3, and those with children know how competitive they can be.

“They’re probably going to want one each, now, too,’’ Jacobson said.

Michael Whitmer can be reached at mwhitmer@globe.com.