Puig, seeded eighth in the event and ranked 48th in the world, outlasted 92nd-placed Watson 6-4, 6-4 in 94 minutes.
Third-seeded Australian Samantha Stosur, the 2011 US Open champion, beat Johanna Larsson of Sweden 6-4, 6-2 to reach the second round of the tournament, carrying a total purse of $235,000.
The singles' field has been diluted by the 11th-hour withdrawal of top-seed Jelena Jankovic of Serbia due to pain in her lower back.
She was injured while losing 6-2, 6-2 to world number one Serena Williams in the final of the China Open in Beijing on Sunday, according to tournament organisers.
Last year's semi-finalist, Misaki Doi of Japan, whipped Spain's Silvia Soler-Espinosa 6-1, 6-2. Her compatriot Kurumi Nara defeated Caroline Garcia of France 6-3, 6-2.
Canadian Eugenie Bouchard, seeded fifth, trounced Varvara Lepchenko of the United States 6-3, 6-0 in the other first-round singles match on the tournament's opening day at the Utsubo Tennis Centre in Osaka.
The 21-year-old Watson beat Chang Kai-chen of Taiwan in full sets in her first-ever WTA final at the Japan Women's Open last year. She has not won a WTA event since.
Defending champion Heather Watson bows out of Japan Open
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