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 U.S. 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.
Thomas has missed the cut in two of his three Open appearances.