The 34-year-old Teater shot rounds of 64 and 69 to take medalist honors. He has never played in a major championship before taking one of eight spots available under very windy conditions at the par 70 course.
The British Open will be held July 18-21 at Muirfield Golf Club in East Lothian, Scotland.
Wagner shot 134, along with Columbia's Camilo Villegas, who did not play in a single major last year.
Brown who won earlier this year in a PGA Tour event in Puerto Rico. He shot a 135. England's Brian Davis also shot 135.
There was a four-man playoff for the final three spots between Americans Bud Cauley, Luke Guthrie, Columbia's Andres Echavarria and Sweden's Robert Karlsson, who all finished at 136.
Echavarria, seeking his first major championship, was eliminated with a bogey four on the first playoff hole while the others made par 3.
Among those who missed qualifying was young American star Jordan Spieth, who lives nearby and played the course dozens of times as a junior. He was on the verge of qualifying, but put two balls in the water on the par-3 17th hole, taking a 7 and sinking his chances for his first Open Championship.
"It's very disappointing, it stinks," he said.
Former Masters champion Trevor Immelman missed with scores of 68-73-141 as did former Ryder Cup captain Davis Love III, 70-72-142, who put a streak of 26 straight Open Championship appearances in danger.
"There are still some chances for me to get in the Open," Love said, "but I want that streak to continue. I want to keep playing."