We present the first detection of the baryon acoustic oscillations (BAOs) signal obtained using unblinded data collected during the initial 2 months of operations of the Stage-IV ground-based Dark Energy Spectroscopic Instrument (DESI). From a selected sample of 261 291 luminous red galaxies spanning the redshift interval 0.4 < z < 1.1 and covering 1651 square degrees with a 57.9 per cent completeness level, we report a similar to 5 sigma level BAO detection and the measurement of the BAO location at a precision of 1.7 per cent. Using a bright galaxy sample of 109 523 galaxies in the redshift range 0.1 < z < 0.5, over 3677 square degrees with a 50.0 per cent completeness, we also detect the BAO feature at similar to 3 sigma significance with a 2.6 per cent precision. These first BAO measurements represent an important milestone, acting as a quality control on the optimal performance of the complex robotically actuated, fibre-fed DESI spectrograph, as well as an early validation of the DESI spectroscopic pipeline and data management system. Based on these first promising results, we forecast that DESI is on target to achieve a high-significance BAO detection at sub-per cent precision with the completed 5-yr survey data, meeting the top-level science requirements on BAO measurements. This exquisite level of precision will set new standards in cosmology and confirm DESI as the most competitive BAO experiment for the remainder of this decade.