The 28-year-old right-hander, who began the season in the minors before being promoted to Oakland on May 30, gave up two singles in two innings against Texas but was otherwise flawless. He struck out three and started an inning-ending 1-6-3 double play in the eighth before giving way to closer Huston Street.
Ziegler broke the previous mark of 25 innings set by Philadelphia Phillies right-hander George McQuillan in 1907. Ziegler's scoreless streak also is the third-longest by an Oakland reliever.
The Athletics beat the Rangers 6-5.