Bergeron's slap shot late in the first period put the Islanders ahead and he knocked in his second goal in the third period. Both goals came on the power play.
Daniel Alfredsson scored for the Senators, who had won their previous four games.
Ottawa was without high-scoring wing Dany Heatley, sidelined for up to six weeks with a shoulder separation suffered in the Senators' 3-2 win over the Detroit Red Wings Saturday.
Notable deaths of 2014 [PHOTOS]