The victory ended Washington's three-game losing streak and moved the Capitals into a tie with Tampa Bay for first place in the Southeast Division.
Jason Chimera scored Washington's other goal later in the period. Those three goals came in a stretch of 6 minutes, 16 seconds as the Capitals turned a 1-0 deficit into a 3-1 lead.
"They executed better," said Washington coach Bruce Boudreau. "Ottawa was tremendous after two periods in not hardly allowing anything. You keep doing the same thing over and over and eventually, we broke through."
The Capitals seemed to take off after Laich got the game-tying goal at 7:15 of the third period. Carlson scored the game-winner just 45 seconds later, and Chimera added one more at 13:31.
"This year, it seems like we are going to the 30-40 minute mark before we score," Laich said. "Tonight it's even further, but once we got one, the crowd gets into it, and the guys start rolling and (Carlson) makes a great shot, and (Chimera) makes a great shot and we end up with a 3-1 win."
Both teams have had problems scoring lately, and those troubles continued in this game. The Senators scored only 16 goals in their last nine games and were ranked 29th in scoring coming in.
Washington's offensive troubles also continued. The Capitals have scored three goals or fewer in their last 10 games and didn't create many scoring chances until the third period.
The Capitals trailed for most of the game after Mike Fisher gave Ottawa a quick lead with his goal just 72 seconds into the contest. He scored after Washington couldn't clear a loose puck from in front of goalie Michal Neuvirth.
That lead held up into the third period when Alex Ovechkin and Laich combined to help get the tying goal.
Ovechkin hit Erik Karlsson hard in the left corner, and the puck squirted over to teammate Chris Phillips on the other side. Phillips then tried to make a quick pass out of the zone, but Laich intercepted and fired a snap shot past goalie Brian Elliott at 7:15 of the third period.
The Capitals took the lead just 45 seconds later on Carlson's power-play goal. Milan Michalek got called for cross-checking at 7:58, and Carlson needed just two seconds to get the go-ahead goal. Nicklas Backstrom drew the face-off back to Carlson at the right point, and he beat Elliott with a hard slap shot to make it 2-1.
Chimera gave Washington an insurance goal with 6:29 left, getting the kind of break the Capitals haven't seen much of in recent weeks.
Chimera threw a shot towards the goal from the left corner with almost no angle to it, but the puck hit Elliott on his back and riccocheted into the net.
"Sometimes you try to bank it off (the goalie), and that time I did and fortunately it went in," Chimera said. "Nine times out of 10, it doesn't go in. Fortunately it worked out."
Neuvirth made just his second start in goal since Dec. 23 and finished with 22 saves, keeping Washington close until the offense awakened in the third period.
The rookie improved to 3-0-0 against Ottawa this season. The Capitals needed Neuvirth to play well because they started slowly once again.
Washington got only five shots in the first period and didn't create many scoring chances. That's the 15th time in the last 21 games they've either been trailing or tied after 20 minutes.
Ottawa wasted the early lead that it held for much of the game and now has lost eight of nine.
"It has been the same thing for so many of our losses--we've played well, we just can't score on any of our chances," said Sens coach Cory Clouston. "We just couldn't extend the lead, and that was the difference."
Material from wire services and team media was used in this report.
1 - 0 OTT
1 - 1 Tie
2 - 1 WSH
3 - 1 WSH