Trailing 2-1 since the first inning, the Athletics got ninth-inning singles from Yoenis Cespedes and Brandon Inge, putting two runners aboard with two outs against Giants closer Santiago Casilla (1-3).
Norris, a catcher playing in only his third major-league game, then worked the count to 3-2 before slamming a pitch into the left-field stands for his first career homer.
The clout made a winner of Jim Miller (2-0), who hurled 1 1/3 innings of scoreless relief as the A's avoided a three-game sweep at the hands of their Bay Area rivals.
Oakland starter A.J. Griffin pitched well in his major-league debut, holding the Giants to two runs on three hits over six innings.
Buster Posey cracked a two-run homer for San Francisco.
Giants starter Matt Cain allowed only one run on three hits over seven frames.