Cole now has three goals over his past two games.
"It's always nice to be contributing and take some pressure off our top line so they don't have to be there every night for us," Cole said. "It's just two games. There's a long way to go."
"[Lehtonen] made some great saves. He was our best player, by far,” Stars coach Lindy Ruff said.
Radim Vrbata, Zack Kassian and Jannik Hansen scored for the Canucks, who were trailing 5-0 before Vrbata's goal at 3:39 of the second period. Ryan Miller stopped eight of 13 shots before being replaced by Eddie Lack, who made 14 saves.
Vancouver started the game strong and looked poised to take an early lead. But Dallas scored first when Canucks left wing Alexandre Burrows inadvertently knocked a pass from Garbutt into his own net 4:08 into the game.
"That was just great," Garbutt said of his first goal of the season. “It was big jumping out to the lead."
The Canucks had a great chance to even the score at 16:51 of the first when Hansen had a breakaway, but Lehtonen made a pad save on Hansen's wrist shot.
"When you go and make a save like that, that gives you some momentum," Ruff said. "That's the ultimate tradeoff. You get that big save and then you go the other way. It was a situation where they're either coming back or we go the other way and get one. A lot of times you see that, when one big save leads to a goal going the other way."
Dallas added a second goal less than a minute later via the power play when Cole beat Miller with a wrist shot from the high slot at 17:46.
With less than a minute left in the first period, Seguin made it a 3-0 lead with a slap shot from the right circle that beat Miller after a pass from Jamie Benn from behind the net.
"I think the first period was pretty even," Canucks coach Willie Desjardins said. "They just scored on their chances."
That momentum carried over when Benn scored 48 seconds into the second period with a slap shot from the right circle that Miller appeared to be positioned perfectly to deny. But Benn's hard shot careened in off Miller's blocker.
"It hit my blocker and then it hit my knob, right? Like I couldn't do that again if I tried, line up a million pucks. He kind of fanned on it, reached for it, hit my blocker and it was going wide and hits my stick I think," Miller said. "I don’t even know what to say. I haven't done that one in the NHL before. I've done a lot of stupid things but that's a new one."
Dallas added a fifth goal 29 seconds later when Eakin flicked a wrist shot past Miller at 1:17, prompting Desjardins to replace Miller with Lack.
Vancouver got on the board when Vrbata scored from a sharp angle, flipping the puck in from the left side of the net.
But Cole scored again 1:05 later, this time when a pass from Shawn Horcoff deflected in off his skate to make it 6-1.
A tip-in by Kassian with 6:05 remaining in the second made it 6-2.
"That's a huge positive," Stars defenseman Trevor Daley said of Dallas' ability to score in bunches. "That's going to help us win games. On the flipside, we have to be good on the [defensive] side. We got two points tonight. It's a huge two points. We hadn't won at home yet. At the end of the year they don't ask how, they ask how many."
The Canucks looked to have made it a three-goal game early in the third when Henrik Sedin appeared to score at 2:54. But his goal was waved off after officials determined that Daniel Sedin made incidental contact with Lehtonen inside the paint.
"Even if we get the first one, it gets waved off if there's contact," Hansen said of the no-goal call. "But I mean, we get a couple calls going the other way or a couple bounces here and there and this could have been a different game. But again when push comes to shove, we can't put ourselves in a hole like that."
Vancouver did cut it to 6-3 at 6:45 of the third when Hansen scored from the slot while Lehtonen was distracted by a flurry of activity in front of the net.
"I thought we played pretty hard in the third. I didn't think we gave up. That's the only positive," Desjardins said.
The Canucks outshot Dallas 46-28 in the game, but that offered little solace to Vancouver's first-year coach.
"I think when you're down, you always gamble more and when you're gambling more, sometimes you get more shots," Desjardins said.
Vancouver returns to the ice Thursday at the St. Louis Blues; Dallas next plays Friday at the New Jersey Devils.