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The Legacy Survey of Space and Time (LSST) at the Vera C. Rubin Observatory will discover tens of thousands of extragalactic transients each night. The high volume of alerts demands immediate classification of transient types in order to prioritize observational follow-ups before events fade away. We use host galaxy features to classify transients, thereby providing classification upon discovery. In contrast to past work that focused on distinguishing Type Ia and core-collapse supernovae (SNe) using host galaxy features that are not always accessible (e.g., morphology), we determine the relative likelihood across $12$ transient classes based on only 19 host apparent magnitudes and colors from $10$ optical and IR photometric bands. We develop both binary and multiclass classifiers, using kernel density estimation to estimate the underlying distribution of host galaxy properties for each transient class. Even in this pilot study, and ignoring relative differences in transient class frequencies, we distinguish eight transient classes at purities significantly above the 8.3% baseline (based on a classifier that assigns labels uniformly and at random): tidal disruption events ($48\%\pm27\%$, where $\pm$ indicates the 95% confidence limit), SNe Ia-91bg ($32\%\pm18\%$), SNe Ia-91T ($23\%\pm11\%$), SNe Ib ($23\%\pm13\%$), SNe II ($17\%\pm2\%$), SNe IIn ($17\%\pm6\%$), SNe II P ($16\%\pm4\%$), and SNe Ia ($10\%\pm1\%$). We demonstrate that our model is applicable to LSST and estimate that our approach may accurately classify 59% of LSST alerts expected each year for SNe Ia, Ia-91bg, II, Ibc, SLSN-I, and tidal disruption events. Our code and dataset are publically available.