Logan Couture and Brent Burns also scored for San Jose (15-10-1), which played for the first time since Friday and saw its three-game winning streak end.
Stone gave Ottawa a 1-0 lead at 3:42 of the first period on the power play, his ninth goal of the season
Karlsson made it 2-0 at 7:19 with his seventh goal of the season, a wrist shot from above the right circle. The goal gave him 411 career points, and he passed Wade Redden to become Ottawa's all-time scoring leader among defensemen.
Couture cut Ottawa's lead to 2-1 at 8:19 of the second period with a power-play goal. It was his 10th of the season and sixth on the power play.
Burns made it 2-2 with a wrist shot from above the right circle at 6:30 of the third period with the Sharks and Senators playing 4-on-4. Paul Martin had the assist, giving him 300 career points.
The Sharks went 1-for-6 on the power play, while the Senators went 1-for-4.
"Just a weird flow of the game," Burns said. "Killing penalties, power play. We're getting chances, the puck's bouncing everywhere. Just one of those weird games I think. Two of their goals were just crazy bounces."