Robertson recorded his seventh NHL hat trick by scoring three goals in a span of 5:20 to help Dallas build a 4-0 lead.
Roope Hintz scored and had two assists, and Mikko Rantanen had a goal to reach 800 NHL points (324 goals, 476 assists) for the Stars (9-5-1), who have won three of their past five games. Thomas Harley and Wyatt Johnston each had two assists. Jake Oettinger made 20 saves.
Jordan Kyrou had a goal and two assists, and Ryan Leonard and Jacob Chychrun each had a goal and an assist for the Capitals (8-6-0), who have lost two in a row. Dylan Strome had two assists.
Logan Thompson was pulled at 5:43 of the second period after allowing four goals on 11 shots. Charlie Lindgren made 10 saves in relief.
Joel Kiviranta put the Stars up 1-0 at 8:56 of the first period. Radek Faksa hit the right post and the puck slid behind Thompson, where Kiviranta backhanded it into an open net.
Robertson made it 2-0 with a power-play goal just 23 seconds into the second period after he buried a wrist shot from the left face-off circle.
Robertson increased the lead to 3-0 at 3:18, avoiding a check from Chychrun at the right boards and roofed a shot from the top of the right circle over Thompson's blocker.
Robertson completed the hat trick when he put a wrist shot through traffic in off the left post from the point at 5:43.
Kyrou cut the deficit to 4-1 on the power play at 11:00 when he backhanded the rebound off a Cole Hutson shot past Oettinger's right pad.
Rantanen pushed it to 5-1 at 14:50 after his snap shot deflected in off the knob of Lindgren's stick.
With the goal, Rantanen became the fourth Finnish player in NHL history to reach the 800-point milestone, following Teemu Selanne (1,457), Jari Kurri (1,398) and Saku Koivu (832).
Boone Jenner beat Oettinger five-hole with a low wrist shot from the right circle to make it 5-2 at 16:58.
Chychrun one-timed a power-play goal at 19:30 cut it to 5-3.
Hintz put Dallas back up 6-3 at 1:23 of the third period after his intended pass deflected off the left skate of Capitals defenseman Martin Fehervary and popped in over Lindgren.
Anthony Beauvilier's shot hit off Justin Sourdif and then off Oettinger before rolling into the net to bring Washington to within 6-4 at 12:55.
Leonard put a snap shot from the left circle into the top left corner at 18:58 for the 6-5 final.
Stars defenseman Miro Heiskanen left the game early in the second period with an upper-body injury after he was hit into the boards by Sourdif.
