Semin gave Washington a 3-2 lead, scoring on a rebound, after the Capitals blew a two-goal lead with just over four minutes to play.
"It seems like we say it after every game," Kolzig said. "Don't leave our games early. The resolve to get the winning goal says a lot about our team."
Tomas Fleischmann and David Steckel also scored for the Capitals. The victory kept Washington two points behind Southeast Division-leading Carolina.
"It's going to be a big race," Fleischmann said. "Everyday game is a like a playoff game right now."
Goals by Vinny Prospal with 4:11 left and Jan Hlavac 33 seconds later pulled Tampa Bay even at 2. The Lightning, fifth in the Southeast, dropped eight points behind the Hurricanes.
"A tough one," Tampa Bay right wing Martin St. Louis said. "We're not going to quit. We're still in the hunt."
Lightning goalie Johan Holmqvist made 32 saves.
Washington converted one of three power plays during the first when Fleischmann scored on a backhander to give the Capitals a 1-0 lead 16:12 into the game. The Capitals nearly scored on the other two opportunities, but Semin hit the post and Alexander Ovechkin had a shot from the blue line go just wide.
Fans were entertained by end-to-end action during a stretch of uninterrupted play that lasted just under 10 minutes in the second. Kolzig and Holmqvist exchanged several big saves during the run.
"That was the fastest second period I've ever experienced," Kolzig said.
The Lightning continued to apply pressure as the second continued, but Kolzig stopped an in-close try by Brad Richards and Hlavac's shot during a partial breakaway.
"Kolzig gets credit the whole game," Lightning coach John Tortorella said. "He's the difference."
The Lightning outshot Washington 24-12 during the second, including 18-2 over the final 11½ minutes.
Tampa Bay got its first power play at 18:17 of the second, but were unable to score despite taking seven shots.
"We played hard, but we didn't get it done," Tampa Bay center Vincent Lecavalier said.
Steckel made it 2-0 at 13:52 of the third. Four of Steckel's five career goals have come against the Lightning.
- Capitals D Tom Poti missed his second consecutive game because of an undisclosed lower body injury.
- Tampa Bay D Brad Lukowich sat out his third straight game due to an abdominal-groin injury.