The south coast club will pay £300,000 for the 24-year-old New Zealander, equalling their record transfer fee.
Fallon only moved to the Swans from Swindon Town 12 months ago, netting 13 times in 48 appearances for the Welsh side.
Argyle boss Ian Holloway said: “He is a normal person with no airs and graces. He is not arrogant about how good he is and he just wants to do well for the Green Army.
“When you look at some of the chances we have had, we have been crying out for someone like him.
“There is no pressure on this young fella. I want him to enjoy himself and feel part of it, and settle down as quickly as he can.”