Nara, ranked 87th in the world, consistently held her nerve better on the big points as she saved all but two of the nine break points against her, while taking three of her five chances against Puig.
Puerto Rican Puig, who beat defending champion Heather Watson in the opening round, looked the more comfortable of the two during the second set, but in the tie-break it was again Nara who held her nerve better to earn a victory.
Nara will face Polona Hercog in the third round, with Hercog enjoying the luxury of a walkover after Wimbledon runner-up Sabine Lisicki pulled out.
Liscki, who has not made the quarter-finals of any tournament since Wimbledon, had taken a medical time out in her opening match of the tournament against Chaneelle Scheepers, and she was unable to play on Wednesday.
Third seed Sam Stosur came through a straightforward match against Swiss qualifier Belinda Bencic, winning 6-4 6-2.
"I feel good," Stosur, the Osaka champion in 2009, said. "I'm excited to be in the quarter-finals again here and overall I'm very happy with the way I played today."
Stosur will face the winner of the all-Japanese battle between Kimiki Date-Krumm and Misaki Doi, who play on Thursday.
Japan Open Women's Singles Round 2 matches on Wednesday
Kurumi Nara (Japan) beat 8-Monica Puig (Puerto Rico) 7-5 7-6(4)
3-Samantha Stosur (Australia) beat Belinda Bencic (Switzerland) 6-4 6-2
Barbora Zahlavova Strycova (Czech Republic) beat Vania King (U.S.) 2-6 6-3 6-1
5-Eugenie Bouchard (Canada) beat Luksika Kumkhum (Thailand) 6-2 6-2