Vernon Fiddler and Erik Cole also found the back of the net and Kari Lehtonen returned from a three-game absence due to a groin injury to stop 37 shots for the Stars, who are 6-1-0 in their last seven games.
"We obviously know how tight the standings are," said Benn, who also added an assist for his second straight two-point night following a seven-game stretch with just two assists. "We had our game in hand tonight and we got the two huge points that we needed."
Dallas moved into a tie with Detroit for ninth in the Western Conference, two points behind Columbus, which lost to Los Angeles, for the eighth and final playoff spot. The Stars have five games remaining, including one against the Blue Jackets in Dallas on April 25 and the finale against Detroit at home on April 27.
"From here on now, we kind of control our own destiny and every game is big for us," said Eakin, who had just one assist in his previous nine outings. "We're ready to give our all, and scratch and claw every chance we get, and play our game from here on in."
Stars coach Glen Gulutzan was encouraged by Benn's last two games, knowing that a hot Benn will go a long way toward helping Dallas end a four-year playoff drought.
"There's no way around it, we need our big players to be big in these next five games and it's good to see," Gulutzan said. "Confidence is a big thing and getting on a roll is a big thing and it's nice to see your big guy feeling good."
Derek Roy, who was traded from Dallas at the trade deadline on April 2, scored for Vancouver, which had a chance to clinch a playoff spot with a win. The Canucks lead the Northwest Division by four points over Minnesota, which was playing later against San Jose later.
"We just made some mistakes in the third that ended up in the back of our net," Vancouver coach Alain Vigneault said. "They got some momentum and in the third, they obviously made plays and we didn't. The plays that we didn't make ended up in the back of our net."
Cory Schneider made 25 saves for the Canucks, who are 5-2-1 in their last eight outings and concluded a season-long five-game road trip with a 2-2-1 mark.
"It's definitely not good enough, especially down the stretch here," Schneider said. "If you want to win in the playoffs, you've got to protect leads and right now, we're not as good as we should be at it and not what we expect of ourselves. Whether it's me coming up with some big saves in the third to preserve it or us as a group just playing harder, we've got to find a way."
Dallas spotted the Canucks a 1-0 lead, tied it late in the second period and then scored three goals within the first 7:03 of the third. Lehtonen was outstanding in goal along the way.
"It took me a little bit to get comfortable, but I had lots of energy," said Lehtonen, who had started 19 straight games before his injury. "I was just trying to contain that and not be too excited. It was fun to play and things went well, I got lots of bounces and it was nice to get all those goals in the third period, too."
With the teams tied 1-1 heading into the final period, the Stars' offensive assault began with Eakin's seventh of the season just 1:36 into the frame.
Benn recorded his third goal in the last two games at 4:09 when a botched exchange between Alex Burrows and Keith Ballard at the Vancouver blue line allowed Benn to race in on a breakaway.
"It was kind of just a turnover by them, I maybe got a little bit lucky," said Benn, who scored twice in the Stars' 5-2 loss in Chicago Monday night. "I just tried to make a quick shot and it was fortunate enough to go in."
Whitney completed the barrage with his 11th goal at 7:03.
Dallas capped off the triumph with Cole's power play goal with 1:42 left, coming off a wrist shot from the slot that beat Schneider past the blocker, just inside the left post.
"We're 44 games into the season," Schneider said. "Playoffs start in a week and a half. I think we're past the point of learning lessons. I think we need to go out and do it, execute and be the team that we know we can be because I don't think we're showing it for a full 60 minutes."
Vancouver D Kevin Bieksa missed his second game in a row with a groin injury, while fellow blue-liner Christopher Tanev missed his sixth straight with a leg ailment. ... Dallas rookie RW Alex Chiasson, who scored six goals and an assist in his first seven NHL contests after his recall from AHL Texas, sat out due to a shoulder injury. He is listed as day to day. ... The Stars honored former D Sergei Zubov for being part of their 20th anniversary all-time team. Zubov, who played in Dallas from 1996-2009, was a key part of the franchise's 1999 Stanley Cup team.
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DALLAS, TX - APRIL 18: Reilly Smith #18 of the Dallas Stars skates before a game against the Vancouver Canucks at...