In addition to the good suggestions above, here are a few that might work (the top 2 are probably the best fit):
Lunch Queen--Cute, light, funny. Although it doesn't really have chemistry, romance is the main plot line and it ends happy too.
Ordinary People (Asunaro Hakusho)--A little bit old, but really sweet romance story. Some drama, but ends well. Takuya Kimura is an added bonus.
Taisetsu na Koto wa Subete Kimi ga Oshiete Kureta--About love and marriage and everything in between. Pretty angsty, but lovely ending.
Hotaru no Hikari--Hilarious! About falling in love. Not a very mature romance, but happy ending.
Tokyo Friends--This one is short and sweet. It's a friendship drama as much as a romance one, but it's one of my favorites.
Kamisama Mou Sukoshi Dake--Deeply touching and romantic love story. Imo, the ending is happy. Although that's debatable.
Last Friends--Love triangles and dark romance. Happy ending, but you have to wade through a lot of angst first.
Sore wa Totsuzen Arashi no youni--Wistful romance. Really good drama. Happy ending, but not necessarily what you might expect.
Watashi ga Renai Dekinai Riyuu--This one is probably more on the order of a Love Shuffle type show. Romance and the psychology of relationships is the main theme, but it doesn't focus on one love story.
