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.
UPI Sports Results for September 20, 2014
UPI Sports Results for September 19, 2014