Edmonton extended its NHL record for most shootout wins in a season to 14.
Hemsky, the last of Edmonton's shooters in the shootout, beat goalie Pascal Leclaire with a forehand and Oilers goalie Mathieu Garon made it stand with a left pad save on Gilbert Brule to end the game.
It was the seventh win in a row at home for the Oilers (31-30-5), who went over the .500 mark for the first time since the fourth game of the season, something they failed to do in five previous opportunities. Edmonton is seven points behind eighth-place Colorado in the Western Conference playoff race.
Columbus (30-27-10) has lost two of its last three and is four points back of eighth.
Nash scored with just over three minutes remaining in the first period to put the Blue Jackets on the board, beating Garon with a wrist shot from the top of the circle.
It only took the Oilers 41 seconds to answer as Cogliano tipped a shot from fellow rookie Sam Gagner through the legs of Leclaire.
It stayed 1-1 through a scoreless second, with Garon facing the majority of the shots as Columbus outshot Edmonton 24-12 through the second period.
The Blue Jackets regained the lead midway through the third as Nikolai Zherdev made a deft backhand pass from behind the net to Chimera, who tucked the puck between Garon's legs.
The Oilers scored goals just 20 seconds apart a couple minutes later to take the 3-2 lead.
They tied the game on the power play when Hemsky walked in from the point and beat Leclaire on a high shot. They jumped out in front on a bit of a fluke when Staios' point shot was deflected and went high in the air and over Leclaire, who made an attempt to jump up and catch it before it landed in the net behind him.
Tarnstrom scored the game-tying goal against his former team with a shot through the crowd.
Linesman Ryan Galloway took a puck in the face in the first minute of the contest and left the game, bleeding from just below the eye. He was able to return for the second period after getting 10 stitches ... It was the first meeting between the teams since Columbus sent forward Curtis Glencross to Edmonton for Tarnstrom. Glencross has two goals and three assists in his first nine games with Edmonton.