Others who failed to qualify for the final 36 holes included Spaniards Jon Rahm and Sergio Garcia, American Bubba Watson, Japan’s Hideki Matsuyama and Englishman Ian Poulter, as the championship lost a number of the game’s biggest names.
Irishman Padraig Harrington, the champion when the Open was most recently held at Carnoustie in 2007, also bowed out, though not unexpectedly given his recent poor form.
World number three Justin Rose came ever so close to exiting, before sinking a 15-foot birdie putt at the last to advance with nothing to spare.
A month after falling narrowly short at the US Open, where he finished third, world number one Johnson suffered a different kind of disappointment. The par-four 18th proved his nemesis, with a triple-bogey on Thursday, when he hit his tee shot out-of-bounds, followed by a double-bogey on Friday.
He carded 76 and 72 for a six-over total of 148, three strokes too many to advance.
It is the first time Johnson has missed an Open cut since his debut at Turnberry in 2009.
World number two Thomas was sailing along nicely until running up three successive double-bogeys starting at the par-five sixth.
He steadied his leaking vessel, however, and still looked likely to survive the cut, only to miss it by one shot after bogeys at the 16th and 17th holes — and a birdie putt that lipped-out at the last — for a round of 77.
Meanwhile, overnight pacesetter Kevin Kisner and former champion Zach Johnson settled for a share of the clubhouse lead, with Tommy Fleetwood right in contention behind them.
Kisner and Johnson are sharing a house in Carnoustie this week along with several other American stars competing for the Claret Jug, but they did not share the same conditions out on the course.
The 42-year-old Johnson, who won at St Andrews in 2015, survived miserable morning rain as he followed his 69 in the first round with a second-round 67 to move to six under par for the tournament.
He recovered from a bogey at the first hole with five birdies and no more dropped shots after that as he again flourishes at the one major held outside his home country.
“The reverence I have for this championship and specifically that trophy, that Claret Jug, I’m not suggesting that someone doesn’t have a higher reverence for it, but I’d argue with them,” said Johnson, who had made the cut at 11 straight Opens before this one.
“I’m not going to sit here and say, man, I just love playing in the wind and rain because I don’t, but I can do it, you know.”
Kisner does not have the same CV as Johnson, also a former Masters winner, although he did lead last year’s US PGA Championship going into the final day before finishing joint-seventh.
After his stellar opening 66, five under par, Kisner shot a 70 yesterday as the rain disappeared and the sun came out for his afternoon round. But it could have been so much better as he was eight-under standing on the 18th tee.
Rory McIlroy, the 2014 Open champion, is in a strong position too, after going out shortly before 8am and shooting his second successive 69 to go to four-under for the tournament, alongside South Africa’s Erik van Rooyen.
The Northern Irishman, looking for his first major in four years, had four birdies for the day as well as dropped shots at the 13th and 14th holes, and was happy with his effort.
“There is still a long way to go, but I’m in a great position going into the third day,” he said.
If the prospects are good of a sixth straight major being won by an American, it is looking a little less likely that this tournament will finally be the one where Tiger Woods ends his decade-long major drought.
He shot a second straight even-par round of 71, but at least he will be here for the weekend, in contrast to world number one Dustin Johnson, whose underwhelming 72 leaves him six over par for the tournament.
The last world number one to miss the Open cut was England’s Luke Donald at Royal St George’s seven years ago.
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