The Sabres’ lone point came in their home opener with an overtime loss.
Brent Seabrook (1G, 2A) sent a rising wrister from the blue line, and Smith tipped in his first goal of the season with 10:34 left in the first for a 1-0 Blackhawks lead.
Hawks goaltender Corey Crawford made 28 stops in his second win of the season, following a night off.
Hawks defensemen Michael Kostka and Michal Rozsival were also healthy scratches.
Veteran winger Patrick Sharp appeared in his 600th NHL career game for the Hawks, while waiting on the birth of his child.
The Hawks will travel to face the Carolina Hurricanes on Tuesday night at 6:00 p.m. CT (CSN-CH, WGN Radio).