The Blackhawks moved into a tie with Nashville and Detroit for fourth place in the Western Conference at 92 points. The next six teams in the standings are separated by just three points.
Corey Crawford gave up a goal to Jannik Hansen 27 seconds into the contest and then turned aside the remaining 23 shots he saw.
Patrick Kane tied it for the Blackhawks 3:12 into the second period.
The winning goal came when Shaw tipped in a shot by Johnny Oduya from the high slot.
Roberto Luongo allowed two goals on 37 shots in Vancouver's fourth loss in five games.