The 21-year-old shot a final-round 70 at St Andrews to post 11 under and overtake Ireland’s Paul Dunne, who had begun the final round with a three-stroke lead over his rival.
Five amateurs made the cut and with three finishing in the top 12, it was the most competitive challenge for the Silver Medal for some time.
Niebrugge, the 2013 US Amateur Public Links champion, rated it as his biggest achievement so far.
“I would rank it up for sure the top accomplishment I’ve had so far: the Silver Medal in a major championship, especially at St Andrews, is definitely a dream come true,” he said.
“I played in the Masters last year and the John Deere Classic (on the PGA Tour) this last summer and both of those experiences definitely made this a little easier.
“There is nothing like coming down the last hole, though. It’s just an awesome amphitheatre – I was a little nervous.”
Niebrugge will not be turning professional, as so many do after a good performance in a major, as he wants to complete his studies.
That means he will be in contention for September’s Walker Cup.
“I’ve still got one more year left at Oklahoma State,” added the American, who came through final qualifying at Hillside Golf Club in Southport.
“I study entrepreneurship and sports management, basically just typical business classes, with a sports management major.
“If golf doesn’t work out, I’d still like to stay in the golf field and do something business-wise that way.”
He beat compatriot and former number one-ranked amateur Oliver Schniederjans, who is wasting no time in joining the paid ranks and will be teeing it up at this week’s Canadian Open on Thursday having already played in two majors.
“I really wanted these experiences in two majors and I’m absolutely thrilled that I decided to do that,” he said, having made the cut in last month’s US Open.
“I feel like I’m ready to be out here.”
England’s Ashley Chesters could find himself competing against Schniederjans again in September after finishing alongside the American on nine under with a final-round 69.
“I’ve got to go back and defend the European Amateur again,” he said of his immediate plans to win a third successive title in Slovakia in August.
“I’ll probably wait and play the Walker Cup now and hopefully this will do me no harm in getting picked for that and then possibly turn pro after that.”
Dunne finished on six under, having gone into the last round as joint leader in the final group on 12 under, after shooting 78 with Frenchman Romain Langasque two-over after a 74.