Lesbian:
The Handmaiden (2016)
Blue Gate Crossing (2002) (it's a love triangle, and 2 of them end up together, so I guess it's a matter of opinion whether or not it's happy)
Saving Face (2004) (not an Asian movie, but the main characters are Chinese-American)
Gay:
The Wedding Banquet (1993)
Also two more recommendations, but I'm not sure if they have happy endings because I don't remember how they ended:
Tropical Malady (2004)
Happy Together (1997)