Vladimir Guerrero, who signed a free-agent deal with the Rangers in the offseason, had two hits and two RBIs against Zito, who struck out four and walked none in 6 1/3 innings.
Kevin Frandsen hit a sacrifice fly in the ninth to give the Giants the win.
C.J. Wilson, getting an opportunity to join the Texas rotation after working as a reliever, pitched five innings. He gave up three runs and four hits.
"[Pitching coach] Mike [Maddux] made a speech the other day, saying that you can have a rough first inning, get a lot of bad bounces, but what matters is how you finish," said the 29-year-old Wilson, who was 5-6 with a 2.81 ERA and 14 saves in 74 games last season.
"My delivery was pretty good, my change up was pretty good and my cutter was working. I'm proud of my change up and how far it's come. It's keeping guys off-balance."
Wilson said his progress in the remaining two weeks for the spring will be gauged on "going out there and getting extended [more innings]."
He understands the difficult decisions facing team officials about where to put certain pitchers.
"It's like riding a teeter-totter," Wilson said. "You need a good bullpen, but you need good starting pitching.
"I'm just going to do the best I can to produce a lot of innings if I am a starter, or put up a lot of zeros as a reliever."
San Francisco infielder Juan Uribe left after two innings with a tight right hamstring. He hit a two-run double in the first inning. He is listed day to day.
Veteran right-hander Guillermo Mota picked up the win in relief, pitching a scoreless ninth. He struck out two and hit a batter. ... Third baseman Pablo Sandoval might return to the Giants' lineup today against the Chicago White Sox in Glendale, Ariz., after missing a few games with a cut in his leg that occurred when he was spiked by Cleveland catcher Lou Marson on Friday. ... Shortstop Elvis Andrus hit his first home run of the spring. ... Texas is off on Tuesday, although right-hander Scott Feldman will stay on schedule by pitching in a minor league game against the Seattle Mariners.