Richard Zednik, Chris Clark, Matt Pettinger and Alexander Semin scored for the Capitals, who won their second straight after six losses. Jakub Klepis had two assists.
Kolzig, who won his 255th career game, had been 0-9-1 against the Minnesota/Dallas franchise before handing the Stars their season-high third consecutive loss. Dallas had beaten the Capitals five straight times.
Jussi Jokinen had two goals and Stu Barnes also scored for Dallas, which dropped its fourth consecutive road game.
Zednik and Clark tallied 1:52 apart in the first for a 2-0 lead.
Zednik's fourth goal of the year put the Capitals up 1-0 13:05 into the game. No one picked Zednik up along the right wing and he snapped a short pass from Brooks Laich past Mike Smith from the right circle.
With Washington on a power play, Klepis won the puck in a scrum along the boards and fed Clark, who scored from between the circles at 14:47.
Jokinen halved the lead 41 seconds into the second period, flipping the puck past Kolzig from high in the slot.
Washington's Alex Ovechkin was awarded a penalty shot midway through the second after Smith slid his stick toward a wide-open Ovechkin in front of the goal. But Ovechkin lost the handle on his drive to the net, and couldn't unleash a shot.
Pettinger got the margin to 3-1 with 2:07 left in the second, just after Washington killed a penalty. Brian Pothier sprung him on a breakaway and Pettinger sidestepped a defenceman at the blue line before scoring on a low shot.
Semin's power-play goal with 12 seconds remaining in the second made it 4-1. Semin whistled a wrist shot from centre ice that nicked Smith's glove and went by him.
Marty Turco replaced Smith for the third period and the Stars got two goals in the closing minutes.
Barnes cut a tough angle from the left side with 3:21 left and Jokinen converted a rebound with 2:03 remaining.
Notes: Dallas visited Washington for the first time since Nov. 13, 2002. ... Capitals RW Matt Bradley missed the game after returning home to Stittsville, Ont., following the death of his grandfather. ... Washington scratched RW Dainius Zubrus (illness), while Dallas skated without LW Brendan Morrow (groin).