Biohybrid systems integrate living materials with synthetic devices, exploiting their respective advantages to solve challenging engineering problems. One challenge of critical importance to society is detecting and localizing airborne volatile chemicals. Many flying animals depend their ability to detect and locate the source of aerial chemical plumes for finding mates and food sources. A robot with comparable capability could reduce human hazard and drastically improve performance on tasks such as locating disaster survivors, hazardous gas leaks, incipient fires, or explosives. Three advances are needed before they can rival their biological counterparts: 1) a chemical sensor with a much faster response time that nevertheless satisfies the size, weight, and power (SWaP) constraints of flight, 2) a design, sensor suite, and control system that allows it to move toward the source of a plume fully autonomously while navigating obstacles, and 3) the ability to detect the plume with high specificity and sensitivity among the assortment of chemicals that invariably exist in the air. Here we address the first two, introducing a human-safe palm-sized air vehicle equipped with the odor-sensing antenna of an insect, the first odor-sensing biohybrid robot system to fly. Using this sensor along with a suite of additional navigational sensors, as well as passive wind fins, our robot orients upwind and navigates autonomously toward the source of airborne plumes. Our robot is the first flying biohybrid system to successfully perform odor localization in a confined space, and it is able to do so while detecting and avoiding obstacles in its flight path. We show that insect antennae respond more quickly than metal oxide gas sensors, enabling the fastest odor localization ever demonstrated by a flying robot. By using the insect chemosensory apparatus, we anticipate a feasible path toward improved chemical specificity and sensitivity by leveraging recent advances in gene editing.