The veteran goalie, who signed with Edmonton on Nov. 8, hadn't played an NHL game since he was bought out by the Philadelphia Flyers with seven years remaining on his nine-year, $51 million contract.
The goalie change seemed to spark the Oilers, and they scored on Crawford at 8:08 of the second. Schultz skated in from the point and fired a shot from near the face-off dot to make it 4-1.
Shaw nearly had his second goal of the night during a late third-period power play, but it was immediately waved off because the puck was clearly kicked into the net.
With two minutes remaining, however, Shaw did score again when he tipped Nick Leddy's shot past Bryzgalov for his seventh goal.
The Blackhawks will play the fifth game of their seven-game road trip in Calgary on Wednesday. The trip coincides with the circus arriving in Chicago. The Oilers will start a three-game trip in Nashville on Thursday.
NOTES: It was the second of three meetings between the teams this season. The Blackhawks won the first, 5-4 in Chicago on Nov. 10. ... Dubnyk was selected as the NHL's third star of the week. He allowed only one goal in a pair of wins over Columbus and Florida. ... Chicago forward Marian Hossa missed the game to attend to a family matter. Fellow forward Bryan Bickell sat out his third straight game because of a lower-body injury. Blackhawks center Michal Handzus played his first game since Oct. 24, returning from a hand injury.