Norrie knocked over world No510 Nicolas Barrientos 6-2 7-5 at the Pueblo Viejo Country Club in Cota, with the left hander sending down 10 aces and winning nine of his 10 service games over one hour and 21 minutes.
The victory pulled Norrie's team level at the end of a first day which started poorly when Evans was upset by 253rd-ranked Nicolas Mejia.
Britain were handed the nightmare trip after failing to qualify for the quarter-finals last year, and Evans clearly struggled to get to grips with the conditions in the raucous home arena en route to his loss.
Evans is scheduled to partner Neal Skupski against former world number one doubles pair Juan-Sebastian Cabal and Robert Farah in the first of matches on the second and final day on Saturday.
Norrie will then take on Mejia in the second match, with Evans' clash against Barrientos rounding out the team's South American trip.