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The generation of ozone by dielectric barrier discharge (DBD) is widely used for water and wastewater treatment, the control of catalytic reactions, and surface treatment. Recently, a need for compact, effective, and economical ozone and reactive oxygen–nitrogen species (RONS) generators for medical, biological, and agricultural applications has been observed. In this study, a novel hybrid DBD (HDBD) reactor fulfilling such requirements is presented. Its structured high-voltage (HV) electrode allows for the ignition of both the surface and volume microdischarges contributing to plasma generation. A Peltier module cooling of the dielectric barrier, made of alumina, allows for the efficient control of plasma chemistry. The typical electrical power consumption of this device is below 30 W. The operation frequency of the DBD driver oscillating in the auto-resonance mode is from 20 to 40 kHz. The specific energy input (SEI) of the reactor was controlled by the DBD driver input voltage in the range from 10.5 to 18.0 V, the Peltier current from 0 to 4.5 A, the duty cycle of the pulse-width modulated (PWM) power varied from 0 to 100%, and the gas flow from 0.5 to 10 SLM. The operation with oxygen, synthetic air, and compressed dry air (CDA) was characterized. The ultraviolet light (UV) absorption technique was implemented for the measurement of the ozone concentration. The higher harmonics of the discharge current observed in the frequency range of 5 to 50 MHz were used for monitoring the discharge net power.