Koivu scored the winning goal with less than three minutes to play, allowing the Wild to rebound from a 6-2 loss in Montreal on Tuesday.
Milan Michalek scored on the power play as the teams approached the midway point of the third period to pull the Senators even with the Wild, and it looked as if that goal would at least get the Senators to overtime.
A pair of goals in the second period allowed the Wild to take a 3-2 lead into the third, and Heatley played a part in both.
He drew a primary assist on Brodin's goal that tied the game 2-2 exactly three minutes into the period and then jammed in a loose puck while behind the net at 6:05.
Heatley pushed a loose puck to the front of the net and then beat Senators defenceman Erik Karlsson to the puck before banking it off goaltender Craig Anderson's right pad and in, all while standing behind the goal-line.
The Wild had the only power play of the first period but it proved to be fruitful for both sides while Jared Cowen sat in the penalty box for interference.
The Senators opened the scoring with a short-handed goal on a nifty pass from Clarke MacArthur to Turris. With his stick between his legs, MacArthur fed Turris in the slot and his one-timer slowly trickled across the goal-line at 5:54.
But just 59 seconds later Pominville took a pass at the corner of the face-off circle and beat Anderson with a quick shot inside the near post.
Pominville, who leads the Wild with 13 goals, skated in the warm-up but was listed as a scratch on the starting line up before eventually suiting up.
Zibanejad gave the Senators the lead at 11:11 but the celebration was short-lived.
Notes: The Senators played their 800th home game in franchise history, and 676th at the building currently named the Canadian Tire Centre ... The Senators have lost only 15 man-games to injury all season. The Wild have lost 62 ... The Wild recalled forward Jason Zucker from the Iowa Wild of the American Hockey League. Zucker had four goals and eight points in 12 games with Iowa this season ... Minnesota has allowed a league-low 23 five-on-five goals this season ... Matt Kassian, Eric Gryba and Joe Corvo were scratches for the Senators while Torrey Mitchell and Mike Rupp didn't dress for the Wild.