We demonstrate the potential of Euclid 's slitless spectroscopy to discover high-redshift ($z>5$) quasars and their main photometric contaminant, ultracool dwarfs. Sensitive infrared spectroscopy from space is able to efficiently identify both populations, as demonstrated by Euclid Near-Infrared Spectrometer and Photometer Red Grism (NISP ${\rm RG}_{\scriptscriptstyle \rm E}$) spectra of the newly discovered $z=5.404$ quasar EUCL J181530.01$+$652054.0, as well as several ultracool dwarfs in the Euclid Deep Field North and the Euclid Early Release Observation field Abell 2764. The ultracool dwarfs were identified by cross-correlating their spectra with templates. The quasar was identified by its strong and broad ${\rm C \small {III]}}$ and ${\rm Mg {\small II}}$ emission lines in the NISP ${\rm RG}_{\scriptscriptstyle \rm E}$ 1206–1892 nm spectrum, and confirmed through optical spectroscopy from the Large Binocular Telescope. The NISP Blue Grism (NISP ${\rm BG}_{\scriptscriptstyle \rm E}$) 926–1366 nm spectrum confirms ${C {\small IV}}$ and $\rm{C \small {III]}}$ emission. NISP ${\rm RG}_{\scriptscriptstyle \rm E}$ can find bright quasars at $z\approx 5.5$ and $z\gtrsim 7$, redshift ranges that are challenging for photometric selection due to contamination from ultracool dwarfs. EUCL J181530.01$+$652054.0 is a high-excitation, broad absorption line quasar detected at 144 MHz by the LOw-Frequency Array ($L_{\rm 144}=4.0 \times 10^{25}\,$W Hz$^{-1}$). The quasar has a bolometric luminosity of $3\times 10^{12}\, {{\rm L}_{\odot }}{}$ and is powered by a $3.4\times 10^9\, {{\rm M}_{\odot }}$ black hole. The discovery of this bright quasar is noteworthy as fewer than one such object was expected in the $\approx$20 deg$^2$ surveyed. This finding highlights the potential and effectiveness of NISP spectroscopy in identifying rare, luminous high-redshift quasars, previewing the census of these sources that Euclid's slitless spectroscopy will deliver over about $14\, 000\,$deg$^2$ of the sky.