Shaw was credited with deflecting Johnny Oduya's shot from the slot past Roberto Luongo. Oduya fired from between the circles after he trailed in late and took a feed from Dave Bolland on the left boards.
Patrick Kane scored his 20th goal in the second period for Chicago, which has won five straight and is 9-1-1 in its last 11 despite playing without captain Jonathan Toews.
Jannik Hansen scored in regulation for Vancouver — on the first shot of the game — to break a 15-game streak without a goal. The Canucks lost their second straight, and have won just one of their last five.
Canucks leading goal-scorer Daniel Sedin was injured in the game and did not return. He may have been hurt when he was elbowed in the head by Chicago defenceman Duncan Keith late in the first period.
Chicago centre Marcus Kruger also left the game with an upper body injury.
Luongo made 35 saves in his return after backup Cory Schneider had started Vancouver's previous two games. Luongo was 0-3-0 and had given up 16 goals on 103 shots in his last four starts — including one no-decision where he allowed three goals on seven shots in the opening five minutes before being relieved by Schneider.
Chicago's Corey Crawford made 23 saves in his fourth straight start. He has won five straight.
The contest between the Western Conference rivals — their 35th over the past four seasons including regular-season and playoff games — was punctuated with penalties and skirmishes.
The Blackhawks and Canucks have squared off in emotional post-season series each of the past three years.
The teams' mutual dislike reached a boiling point midway through the second period when several wrestling matches erupted in front of the Chicago net. The Canucks appeared to be targeting Keith, who eventually tussled with Alexandre Burrows. Both players received roughing and misconduct penalties.
Hansen scored the only goal of the first period just 27 seconds in.
After gaining a step on Oduya, Hansen beat Crawford from the right circle. Crawford misplayed the rising shot, giving Hansen a wide gap on the stick side after dropping too quickly to the ice.
Although the Canucks had all three power plays in the first, Chicago outshot Vancouver 12-6. Luongo was sharp in the opening 20 minutes, stopping several tough chances, including a point-blank deflection by Marian Hossa midway through the period.
Kane made it 1-1 at 3:12 of the second to cap a strong shift and crisp passing play with linemates Hossa and Andrew Brunette. After Brunette fed Hossa behind then net, Hossa feathered a centring pass to Kane, who lofted a shot past a diving Luongo from the slot.
Kane scored 5-on-5, but then the Blackhawks failed on five power-play chances in the second.
Crawford had to be sharp in the third as the Canucks began to dominate.
After taking just nine shots through two periods, Vancouver had 14 in the third. Several were tough, close-in chances.
NOTES: Chicago captain Jonathan Toews missed his 15th game with a concussion, but has resumed skating without contact. Chicago RW Jamal Mayers was ill and sat out. His spot was taken by C Brendan Morisson, who had been a healthy scratch the last 10 games. . Vancouver D Sami Salo returned to the lineup after taking Monday's 2-0 loss in Minnesota off as a planned rest. Also back for Vancouver after being a healthy scratch in Minnesota was LW Mason Raymond. Canucks D Aaron Rome sat out with an undisclosed illness.