Cheechoo scored the first power-play goal midway through the second period to break a 1-1 tie.
With OKC in a five-on-three advantage, Cheechoo scored his second goal with 29 seconds left in the period. C.J. Stretch scored another power play goal with :01 left for a commanding 4-1 lead.
Cheechoo, a 32-year-old veteran who led the NHL in goals seven years ago, signed with the Barons after the NHL lockout ended. OKC needed a scorer after four young Oilers stars returned to Edmonton.
The Barons are 9-0 when Cheechoo scores a goal and are 14-1 when he scores a point.
“We want to ride that lucky (stat) all the way through,” Cheechoo said. “I guess I just have to keep up my part and keep scoring.”
VandeVelde, who scored back-to-back, game-winning overtime goals last year against San Antonio in a second-round playoff series, scored his fourth and fifth goals of the season.
“Vandy was a big part of our success last year and down the stretch this year,” Nelson said. “You like to see a guy who works the trenches hard, and does a good job with the penalty kill get rewarded. Hopefully he's getting hot at the right time.”