We observed the interstellar comet 2I/Borisov using the Neil Gehrels Swift Observatory's Ultraviolet/Optical Telescope. We obtained images of the OH gas and dust surrounding the nucleus at six epochs spaced before and after perihelion (-2.56 to 2.54 au). Water production rates increased steadily before perihelion from (7.0 ± 1.5) × 1026 molecules s-1 on 2019 November 1, to (10.7 ± 1.2) × 1026 molecules s-1 on December 1. This rate of increase in water production rate is quicker than that of most dynamically new comets and at the slower end of the wide range of Jupiter-family comets. After perihelion, the water production rate decreased to (4.9 ± 0.9) × 1026 molecules s-1 on December 21, which is much more rapidly than that of all previously observed comets. Our sublimation model constrains the minimum radius of the nucleus to 0.37 km, and indicates an active fraction of at least 55% of the surface. A(0)fρ calculations show a variation between 57.5 and 105.6 cm with a slight trend peaking before the perihelion, lower than previous and concurrent published values. The observations confirm that 2I/Borisov is carbon-chain depleted and enriched in NH2 relative to water.