Miller stopped all 14 shots he faced in the final period while turning in his busiest performance of the season for the Sabres, who won their second straight following a three-game skid.
Vanek, the NHL's leader in points and goals, scored in the first period and added an assist. Alexander Sulzer put in the go-ahead goal in the final frame for Buffalo.
Christian Ehrhoff added a tally and Cody Hodgson notched a pair of assists for the Sabres.
It was New York's fourth straight loss.
Celebrity Families of 2014 [PHOTOS]