1
general::
Phrase(s): go (someone) one better [and] do someone one better
to do something superior to what someone else has done; to top someone. • That was a great joke, but I can go you one better. • Your last song was beautifully sung, but Mary can do you one better.
McGrawhill's American Idioms And Phrasal Verbs
2
general::
Outdo or surpass someone, as in He went one better than his teacher and came up with five more famous scientists. This expression originated in gambling games, where it meant "to offer a higher stake than one's opponent." [Mid-1800s]
American Heritage Idioms