Smith suffered a dislocated thumb in a 4-3 overtime victory over the Colorado Avalanche Friday in Stockholm, Sweden.
Through 12 games this season Smith had six assists. The 29-year-old signed a four-year contract extension worth $13 million in January.
The Sens and Avs play the second leg of their two-game European series Saturday afternoon.
Nick Paul will draw into the lineup to replace Smith and Mike Condon will get the start in net ahead of Craig Anderson.