Santiago Casilla (1-0) threw a scoreless seventh for the win, and has yet to allow a run in his 16 1-3 innings.
Rangers' starter Scott Feldman (0-1) took a three-hit shutout into the seventh inning before allowing a sun-aided double to Frank Thomas and Cust's third homer in four games.
Feldman, who grew up just across the bay from the Oakland Coliseum, made his second career start in front of supportive fans and received a nice ovation when he was taken out of the contest. He gave up three runs and six hits in a career-high 6 1-3 innings, walking one and striking out a career-high five batters.
A's starter Greg Smith gave up German Duran's first major-league home run leading off the third and then retired 12 of the final 14 hitters he faced. Smith gave up one run and three hits. He walked two and struck out a career-high 10.
Huston Street pitched the ninth for his ninth save in 10 chances.
With the exception of Duran's home run, neither team got a runner past second base through the first six innings.
Leading off the seventh, Thomas lifted a high fly ball into center field that Josh Hamilton seemingly had in his sights. At the last second he tried to make a grab for the ball, which fell just behind him.
Cust, who extended his hitting streak to seven games, drilled a 1-0 delivery into the left field stands to put Oakland ahead.
After an out, Bobby Crosby doubled down the left field line, ending Feldman's day. Frank Francisco gave up a run-scoring single to Ryan Sweeney before getting out of the inning.
The Rangers' 3-4-5 hitters combined to go 0-for-12 with 10 strikeouts.