Corey Crawford made 17 saves for the Blackhawks (25-24-8), who are 0-3-2 in their past five games.
"We're just happy about getting our effort back to where it needs to be, and we'll go from there," captain Jonathan Toews said. "Unfortunately, we couldn't find ways to find the back of the net and open the game up a little."
After an hourlong ceremony to retire the No. 22 and No. 33 jerseys of Daniel and Henrik Sedin, it was the Blackhawks, coming off a 5-3 loss to the Edmonton Oilers on Tuesday, who controlled play early. They had 13 shots on goal before the Canucks got their first at 9:56.
Chicago was penalized for an unsuccessful challenge of a goaltender interference call that resulted in a goal being waved off at 13:58 of the first, and Vancouver capitalized on the ensuing power play.
Sutter one-touched a pass from the goal line to Bo Horvat at the left hash marks for a one-timer over Crawford's glove to make it 1-0 at 15:33.
"We felt in the moment that [Brandon] Saad didn't prevent him from making a save and felt like we had a decent chance of getting a goal," coach Jeremy Colliton said. "We feel pretty good about our penalty kill, so it was worth taking a shot. Obviously, it didn't work out."