Puig, 19, won her first-ever grass court match 6-3, 6-2 to advance to the second round.
Errani has reached the semi-finals in three of the past five Grand Slam tournaments, and finished runner up at last year's French Open, but was only able to come up with 13 winners to Puig's 38 in Monday's match.
Puig's stunning victory sent home just the third top-five women's player in the first round since 2001.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys