Bradford piled up 299 yards on 27-of-38 passing efficiency, leading the Rams (1-0) to scores on three of their last four series to erase a 24-13 second-half deficit.
Cook, playing in his first NFL game, amassed 141 yards on seven catches. His 25-yard grab with 40 seconds remaining set up Greg Zuerlein's go-ahead field goal -- his fourth boot of the game.
Carson Palmer went 26-of-40 passing for 327 yards in his debut for the Cardinals (0-1), but also was intercepted once and lost a key fumble.
Larry Fitzgerald made eight catches for 80 yards and a pair of touchdowns as Arizona lost in its first game under new Coach Bruce Arian.