Stoll scored from the left wing after a turnover at Vancouver blue line. The forward skated in on a 2-on-1, but took the shot himself, picking the top-left corner above Schneider's blocker.
Brad Richardson tied it for Los Angeles at 3:21 of the third period, and Jonathan Quick made 26 saves.
• RESULTS: Kings 2, Canucks 1
Henrik Sedin opened the scoring for Vancouver with a power-play goal in the first period.
Schneider made 35 saves in his third consecutive start after Roberto Luongo lost the first two games.
The Kings will play the second-seeded St. Louis Blues in the second round.
The Canucks dropped out in the first round after leading the NHL in regular-season points for the second consecutive year. Last season, they lost to Boston in Game 7 of the Stanley Cup Final.
Vancouver's new second line of Ryan Kesler, Alex Burrows and Max Lapierre made a quick impression. Burrows fed Kesler from behind the net about a 90 seconds in, but Quick stopped the center's shot from the slot.
The Canucks then ran into early penalty trouble and didn't get another shot for six minutes before Lapierre put one on Quick. Vancouver killed penalties to Dan Hamhuis and Sedin, but couldn't get its power play going on its first advantage.
However, with Sedin double-shifting, the Canucks capitalized on the second when the Vancouver captain put in a cross-ice pass from twin brother Daniel Sedin with 5:56 left in the period.
The opportunity came after Hamhuis kept Mike Rrichards' clearing attempt in at the blue line.
Schneider preserved the lead when he stopped Anze Kopitar on a breakaway in the dying seconds of the first period. Kopitar put the rebound off the post as time expired.
Both goaltenders stole the show in the second period as neither team could score.
In the early going, Schneider stopped Kopitar's first shot and Dustin Brown on the rebound as the Kings outshot the Canucks 6-0 in the first 4:11.
Later, Schneider robbed Richards on a rebound, snaring the puck with his catching glove. With just over two minutes left in the second, Quick stymied Daniel Sedin on a breakaway, lowering his right pad to block a snap shot. Sedin slammed his stick against the glass in frustration as he went to the bench.
Richardson drew the Kings even in the third, tapping in a pass from Drew Doughty. Doughty deked and circled around the Canucks' Keith Ballard and passed the puck back to Richardson from the end line. It was the first goal of the series for Richardson, who missed the first three games while recovering from an appendectomy.
Notes: The Canucks made just one lineup change, inserting winger Dale Weise in place of Zack Kassian, who was the principal acquisition in the trade that sent Cody Hodgsonto Buffalo at the Feb. 27 deadline. … Vancouver defenseman Sami Salo played his 100th career playoff game.