San Jose would answer on the game's first power play as Kevin Lebanc fired a quick rebound off the post off of Corey Crawford's glove and in from a low angle at the bottom of the circle, evening the game 1-1 at 11:30 of the first.
Shaw, scored his first goal since re-joining Chicago at 13:22 of the first. Shaw set a screen in front of Sharks goalie Aaron Dell on an Olli Maatta shot from the point, then picked up the rebound at the top of the crease and slotted a quick wrister home for a 2-1 lead.
The Sharks answered in the final seconds of the period as Patrick Marleau tipped home an Erik Karlsson shot on the man advantage at 19:43 of the first, tying the game 2-2.
In the second period, Shaw found the net again off a feed from Alex DeBrincat after the 21-year-old forced a turnover on a forecheck. The Cat fed the incoming Mutt at the top of the circle, where Shaw fired another quick chance past Dell for a 3-2 lead at 9:27 of the second.
The visitors would answer 33 seconds later as Brent Burns scored his second of the year, firing a high saucer pass across the top of the crease that deflected off of Connor Murphy's glove and past Crawford to deadlock the game again, 3-3.
Dylan Strome scored his first of the year at 12:14 off a give-and-go with Patrick Kane on an odd-man rush. Strome fed Kane in the high slot, before putting a one-timer past Dell on the return feed from Chicago's early-season scoring leader for a 4-3 lead.
For the fourth time on the night, the Sharks answered as Marleau scored his second late-period goal of the game at 17:46 of the middle frame. The veteran cleaned up the rebound from a Timo Meier shot from the point, slotting a shot under the outstretched stick of Crawford.
San Jose took their first lead of the night early in the third as Barclay Goodrow found space in front and buried a Joe Thornton feed from behind the goal for a 5-4 advantage at 4:47.
UP NEXT: Chicago hosts Winnipeg on Saturday, Oct. 12 at 6 p.m. (TICKETS)