The Stars began the evening four points behind Columbus for the eighth and final playoff spot in the Western Conference and the win moved Dallas into a tie with Detroit for the ninth position.
Dallas has won six out of seven in a late-season surge that has given them a chance at the postseason.
It was a 1-1 game heading into the third period, but Eakin, Jamie Benn and Ray Whitney all scored in the first 7:03 of the stanza to put the Stars in control. Erik Cole added a power-play goal with 1:42 left in the period.
Vancouver, which has the lead in the Northwest Division, missed a chance to clinch a playoff spot.
Corey Schneider allowed five goals on 30 shots.