The Senators traded Elliott to Colorado last year, and he joined the Blues as a free agent before the current season.
In his first game back in Ottawa since the deal, Elliott provided the difference as St. Louis improved to 10-0-2 against Eastern Conference teams.
Craig Anderson -- who moved from Colorado to Ottawa in the Elliott trade -- fared poorly and was pulled after yielding two goals on four shots.
Alex Auld relieved and finished with 13 saves as the Senators dropped their seventh straight.
Daniel Alfredsson got Ottawa's only goal.