The Sharks tied it 1-1 at 7:51 of the first period when Evander Kane got the puck in the neutral zone, skated around defenseman Josh Manson, and beat Gibson with a wrist shot.
Tomas Hertl put the Sharks ahead 2-1 with an unassisted, shorthanded goal at 17:14 of the second period, but Ducks forward Rickard Rakell took a cross-crease pass from Jakob Silfverberg and scored a power-play goal at 18:00 to tie it 2-2.
"That was a big momentum swing," Silfverberg said. "Their crowd was really going crazy there. It's tough to lose one on a shorthanded (goal). It was a big goal to get us back in the right direction."
Silfverberg (three assists) and Rakell (one goal, two assists) each had three points.
Karlsson, a two-time winner of the Norris Trophy as the best defenseman in the NHL (2012, 2015), was acquired in a trade from the Ottawa Senators on Sept. 13.
"I thought he was good," DeBoer said. "Like I said, I thought guys came in and tried to execute the game plan. We did a lot of good things. We had them at, we gave up 10 or 12 shots through most of the game. Created a lot of chances ourselves. Didn't finish, didn't extend the lead when we had opportunities to, and that's what happens."