Midway through overtime, Semin collected a loose puck in the right circle and beat goaltender Marty Turco in the upper left corner. Semin was mobbed by teammates after his seventh goal of the season.
The Caps went back in front 5-4 when Sloan, playing in his third NHL game, scored on a rebound at 2:22 of the final period.
Sean Avery scored his first goal for the Stars and added an assist. Dallas signed Avery to a four-year, US$14-million free-agent contract in the off-season, and he was off to a slow start with two assists in his first eight games.
Rookie Fabian Brunnstrom added his fifth goal of the season for the Stars.
Reigning league MVP Alex Ovechkin, a 65-goal scorer for the Caps last season, went without a goal for the sixth straight game.
Avery scored from the left circle off a long rebound midway through the opening period to put the Stars in front.
Fedorov tied it at 1 on the power play, firing home a loose puck from close range with 7:22 left in the first period.
Fleischmann struck from the inside edge of the left circle to give Washington a 2-1 lead at with three minutes remaining in the first period.
Dallas tied it at 2 with 9.6 seconds left in the first when Avery's backhand pass set up Brunnstrom for a power-play goal.
Alexander Semin's pass sent Fedorov in alone, and he beat Turco with a backhander to give Washington a 3-2 edge with 8:37 left in the second.
Fleishmann made it 4-2, intercepting Turco's clearing attempt and firing into the upper right corner with just under four minutes left in the second.
Notes: The Caps were making their first visit to Dallas since Jan. 12, 2006. The teams did not meet last season. ...Dallas is 9-1-2 all-time against Washington at home. ...The Stars, expected to be one of the league's top defensive teams, have given up four or more goals in six of nine games.