Turco stopped 21 shots in his third career playoff shutout - all in this first-round Western Conference series.
For the first time in their history, the Stars have come back from a 3-1 series deficit to force a Game 7. They get to play it Monday night.
Vancouver, which has rallied from 3-1 deficits three times to win series, is also one of the 20 NHL teams who have blown that big lead. The Canucks did that four years ago against Minnesota.
Modano's first goal of the playoffs - and the first by any of the Stars' top five goal-scorers in the regular season - came on a 5-on-3 power play only 3:05 into the game. Jeff Halpern added an unassisted goal with 12:38 left after Vancouver turned the puck over in its zone.
The goal by Modano was a huge lift for the Stars, who had failed to capitalize on other 5-on-3 chances in two losses this series.
Dallas had a longer two-man advantage in the second period of Game 4, but went 1:55 without scoring in that 2-1 loss. In the series opener Vancouver won late in the fourth overtime, the Stars couldn't end the game during a 5-on-3 power play for 39 seconds early in the second OT.
With goalie Roberto Luongo screened by Jere Lehtinen in front of the net, Modano took a pass from Sergei Zubov and shot from the slot between the circles. The puck dinged the right post and went into the net.
Only seconds earlier after Dallas got the two-man advantage, Luongo denied Modano on a shot that almost went in. The puck was between the goal line and the goalie, who was on his backside, when Luongo reached back with his left arm and swiped the puck under him.
Luongo stopped 27 of 29 shots.
Dallas ended a mind-boggling six-game home playoff losing streak, one short of the NHL record.