The 20-year-old Spanish wild card beat the seven-time Grand Slam champion 6-2, 6-4 in just under one hour and 45 minutes on court.
Ranked 479th in the world, Williams has now lost all seven matches she has played this season. It was her first match on clay since Roland Garros five years ago.
Every game was closely contested in a first set marked by numerous unforced errors due to the wind blowing across the Estadio Manolo Santana but the world number 140, playing in her first 1000-level event, was the more consistent of the two.
Williams got off to a better start in the second set, taking a 3-0 lead, before Quevedo began to fight back and the match was then interrupted by rain while the roof over the centre court was closed.
Quevedo won five consecutive games to seal her passage to the second round.