Defenceman Lasse Kukkonen scored in the second period to help the Blackhawks win their second straight following an eight-game (0-7-1) losing streak.
Boucher, who entered the game with an 0-7-1 record, has 13 career shutouts.
Columbus' Fredrik Norrena made 18 saves. The Blue Jackets have lost two in row and are 1-3-0 in their last four.
Thanks to three power-play chances, Columbus dominated territorially in the scoreless first and outshot Chicago 10-4.
The Blackhawks took the game's first five penalties and were forced to kill two brief 5-on-3 disadvantages early in the second.
Kukkonen scored the only goal at 7:04 of the second.
He fired a shot through a screen from between the circles after taking a centring pass from Bryan Smolinski. Chicago's Martin Lapointe jumped straight into the air to allow Kukonnen's low drive to reach the net and slip between Norrena's pads.
Chicago forward Rene Bourque left the game late in the second period after being cut in the neck with by the Columbus right wing Nikolai Zherdev's skate blade during a pileup in the Blackhawks' crease. According to team officials, Bourque was in stable condition when he was taken to The University of Chicago Hospital.
Notes: Top Blackhawks goalie Nikolai Khabibulin, sidelined since Oct. 21 by a broken finger, could return Thursday at Phoenix as Chicago begins a five-game western road trip. ... Chicago F Martin Halvat, out since Oct. 20 with a sprained ankle, has resumed light skating but may not return until December. ... Blackhawks LW Tony Salmelainen missed his fifth game because if a separated shoulder.