Kane, the NHL's leading scorer with 46 points, had the second assist on Duncan Keith's power-play goal midway through the first.
"We're all having fun watching it, enjoying the different fashions that he's been able to keep moving forward," Chicago coach Joel Quenneville said.
Kane could have had more in this one, but the assist was his only mark on the score sheet -- and he wasn't exactly sure what happened on the play.
"I think I passed it and it went in 10 or 15 seconds later," Kane said. "I didn't know I had an assist until they announced it. I was hoping they didn't get it wrong and they've got to call it back."
Kane surpassed Sidney Crosby's 25-game streak, set in 2010-11, for the longest in the league since 1992-93. The Blackhawks' right wing has 16 goals and 24 assists during his run, and a point in 29 of Chicago's 31 games.
Ryan Miller had 26 saves for the Canucks.
Rasmussen made it 2-0 with 8:24 left in the third and Shaw had an empty-netter with 31 seconds remaining. With Miller back in the net, Mashinter completed the scoring with 10 seconds left with his first NHL goal.
"He's been awesome," Keith said. "He's made huge saves when we've needed him, and when he's going like that, it allows us to make more mistakes and play with a little more confidence."
Crawford's shutout was the 16th of his career and followed a 25-save effort in a 2-0 win against Winnipeg on Friday. For the second straight time, the goalie credited his team's perfect penalty killing -- 3 for 3 against Vancouver -- and shot blocking (15).
"Our team is starting to come around," Crawford said. "Our PK is really strong, and it seems like we're just getting better and better here."
The Canucks opened a six-game trip on Sunday that continues to Minnesota, Philadelphia, Detroit, Florida and Tampa Bay.
Keith opened the scoring during a Chicago man advantage at 11:51 of the first. After taking Artemi Panarin's feed, Keith one-timed a wobbling shot from the left circle with Chicago's Artem Anisimov and Vancouver's Matt Bartkowski in front and screening Miller.
The game remained 1-0 until the late in the third as Crawford and Miller hooked up in a duel.
"I thought our guys had a good desire tonight," Canucks coach Willie Desjardins said. "I thought we wanted the game. We did everything we could to win and we just came out on the wrong end of it."
Crawford was especially sharp in the second period. He came up with a quick pad save on Daniel Sedin's tip-in attempt during a power play about five minutes in, then a lunging glove save on Chris Higgins' rebound midway through.
Miller matched Crawford, making tough stops late in the frame including a blocker save on Panarin's one-timer from the left circle with 5:00 left.
Midway through the third, Crawford dropped to make a pad save on Sven Baertschi's one-timer from the slot.
Rasmussen scored his second goal in four games since being recalled from Rockford of the AHL at 11:36 of the third. He tapped the puck in after Bryan Bickell's shot from the slot trickled off Miller's glove.
Crawford's shutout was preserved when an apparent power-play goal by Henrik Sedin with 6:30 left was disallowed. Sedin fired the puck in after his brother Daniel directed the puck to him off his glove.
The Canucks played their first game without D Dan Hamhuis, who suffered a gruesome facial fracture last Wednesday when he was hit in the mouth by a shot by the New York Rangers' Dan Boyle. ... Chicago F Marian Hossa played in his 1,200th caeer game and D Niklas Hjalmarsson in his 500th. ... Vancouver F Radim Vrbata skated in his 900th game. ... Bickell's assist on Rasmussen's assist was his first point in 12 games this season.