Panarin, who hit the post with 1:15 left in the third period, also had an assist. Pierre-Luc Dubois scored, and Sergei Bobrovsky made 42 saves for Columbus.
The Capitals took a 1-0 lead when Wilson scored at 5:52 of the second period, and thought they added to it 2:17 later, but Brett Connolly's goal was challenged and it was ruled he was offside.
Wilson scored with a redirection of a shot by Matt Niskanen to give Washington the first goal in all three games.
Dubois made it 1-1 at 11:18 of the second period with his first NHL playoff goal.
Washington needed 28 seconds of a two-man advantage to make it 2-1 at 14:43. Carlson scored his first goal, his seventh point of the series, with a shot from the top of the left circle after a pass from Nicklas Backstrom. Alex Ovechkin, playing his 100th Stanley Cup Playoff game, had an assist.
Panarin broke out on a two-man rush with Cam Atkinson and one-timed a pass at the lower left circle for his second goal of the series after he won Game 1 in overtime.
The Capitals killed a Blue Jackets power play in the first overtime after Carlson tripped Werenski at 15:42. Atkinson hit the crossbar, the fourth time the Blue Jackets hit a post.