Context. The recurrent nova RS Ophiuchi (RS Oph) underwent a thermonuclear eruption in August 2021. In this event, RS Oph was detected by the High Energy Stereoscopic System (H.E.S.S.), the Major Atmospheric Gamma Imaging Cherenkov (MAGIC), and the first Large-Sized Telescope (LST-1) of the future Cherenkov Telescope Array Observatory (CTAO) at very-high gamma-ray energies above 100 GeV. This means that novae are a new class of very-high-energy (VHE) gamma-ray emitters. Aims. We report the analysis of the RS Oph observations with LST-1. We constrain the particle population that causes the observed emission in hadronic and leptonic scenarios. Additionally, we study the prospects of detecting further novae using LST-1 and the upcoming LST array of CTAO-North. Methods. We conducted target-of-opportunity observations with LST-1 from the first day of this nova event. The data were analysed in the framework of cta-lstchain and Gammapy, the official CTAO-LST reconstruction and analysis packages. One-zone hadronic and leptonic models were considered to model the gamma-ray emission of RS Oph using the spectral information from Fermi-LAT and LST-1, together with public data from the MAGIC and H.E.S.S. telescopes. Results. RS Oph was detected at 6.6σ with LST-1 in the first 6.35 hours of observations following the eruption. The hadronic scenario is preferred over the leptonic scenario considering a proton energy spectrum with a power-law model with an exponential cutoff whose position increases from (0.26 ± 0.08) TeV on day 1 up to (1.6 ± 0.6) TeV on day 4 after the eruption. The deep sensitivity and low energy threshold of the LST-1/LST array will allow us to detect faint novae and increase their discovery rate.