In 1971, Brinkman was part of an eight-player trade which sent himself, third baseman Aurelio Rodríguez and pitchers Joe Coleman and Jim Hannan from the Senators to the Detroit Tigers in exchange for Denny McLain, Don Wert, Elliott Maddox, and Norm McRae. He had his best season defensively in 1972. Playing in all of the Tigers' 156 games, he set American League fielding records for shortstops with the most consecutive games without an error (72), most consecutive chances without an error (331), fewest errors in 150 games or more (7) and the highest fielding percentage in 150 games or more (.990). He also produced a career-high 49 runs batted in, helping the Tigers clinch the American League Eastern Division championship by a half game over the Boston Red Sox. Brinkman only appeared in one game of the 1972 American League Championship Series before he was ruled out for the rest of the season due to a ruptured disc in his lower back. The Tigers lost the championship series to the eventual world champions, the Oakland Athletics, in five games. Brinkman's efforts during the regular season earned him the 1972 Gold Glove Award and he was named the recipient of the "Tiger of the Year" award by the Detroit baseball writers. He also finished ninth in voting for the 1972 American League Most Valuable Player Award.