European captain Clarke’s much-anticipated picks for the biennial match against the United States went instead to England’s Lee Westwood, German Martin Kaymer and Belgian Thomas Pieters.
Knox was overlooked despite sitting 20th in the world, seventh in the FedEx Cup and winning twice in the last 10 months.
The 31-year-old would have been among the nine automatic qualifiers if the points from his WGC-HSBC Champions victory in China last November had counted.
Even then, Knox looked as though he was set to secure one of Clarke’s picks for the European team when he won the Travelers Championship on the PGA Tour in the US three weeks ago.
However, a late thrust from rising European star Pieters, who won the Made in Denmark tournament on Sunday to back up fourth place in the Olympics and second in the Czech Masters, earned him the nod ahead of both the Scot and former world No 1 Luke Donald.
The 24-year-old takes the number of rookies in Clarke’s team to six, joining Masters champion Danny Willett, Rafa Cabrera Bello, Andy Sullivan and Chris Wood.
Westwood’s selection means he will be making a 10th Ryder Cup appearance while two-time major champion Kaymer is set for his fourth match in a row, having secured victory at Medinah four years ago.
Open champion Henrik Stenson, Rory McIlroy, Sergio Garcia and Olympic gold medallist Justin Rose complete the 12-strong side for the 41st Ryder Cup matches from 30 September-2 October in Minnesota.
“It was in incredibly tough decision and I’ve not slept a lot,” said Clarke of having to pick between Knox and Pieters.
“Having to phone Russell yesterday was one probably one of the toughest calls I’ve ever made.
“Some people will think it’s a mistake having Thomas on the team but I looked at him and Russell side-by-side and I found it impossible to leave him off the team.”
On Kaymer and Westwood, he added: “They were two pretty obvious choices because they bring a wealth of experience.”
A delighted Pieters said of the victory that secured his pick: “To win a tournament at that point, with a little bit of pressure on me, was very satisfying.
“I just like to let the golf do the talking and I guess I just had to step it up a little bit at the end.
“It’s pretty cool, to be honest. It has sunk in and I just can’t wait to get on that plane and just be with the team the whole week.”