Elliott limited the Hawks to a shorthanded breakaway goal by Marian Hossa but was otherwise perfect in leading Ottawa to its fourth consecutive victory -- and first win against Chicago since 2003.
Cristobal Huet allowed four goals on 18 shots.
Smith was the first Senator to beat Huet, scoring off a turnover at 6:51 of the first period. Shannon made it 2-0 at 12:31 of the first period.
Hossa cut the deficit in half at 6:18 of the second with a breakaway slap shot.
Regin bumped the lead back to two goals with 1:55 left in the second period. Fisher added an insurance goal with 3:15 remaining in the game.