The Cherenkov Telescope Array (CTA) will be the next generation gamma-ray observatory, which will consist of three kinds of telescopes of different sizes. Among those, the Large Size Telescope (LST) will be the most sensitive in the low energy range starting from 20~GeV. The prototype LST (LST-1) proposed for CTA was inaugurated in October 2018 in the northern hemisphere site, La Palma (Spain), and is currently in its commissioning phase. MAGIC is a system of two gamma-ray Cherenkov telescopes of the current generation, located approximately 100~m away from LST-1, that have been operating in stereoscopic mode since 2009. Since LST-1 and MAGIC can observe the same air shower events, we can compare the brightness of showers, estimated energies of gamma rays, and other parameters event by event, which can be used to cross-calibrate the telescopes. Ultimately, by performing combined analyses of the events triggering the three telescopes, we can reconstruct the shower geometry more accurately, leading to better energy and angular resolutions, and a better discrimination of the background showers initiated by cosmic rays. For that purpose, as part of the commissioning of LST-1, we performed joint observations of established gamma-ray sources with LST-1 and MAGIC. Also, we have developed Monte Carlo simulations for such joint observations and an analysis pipeline which finds event coincidence in the offline analysis based on their timestamps. In this work, we present the first detection of an astronomical source, the Crab Nebula, with combined observation of LST-1 and MAGIC. Moreover, we show results of the inter-telescope cross-calibration obtained using Crab Nebula data taken during joint observations with LST-1 and MAGIC.