All the offense was produced in the first period as the game was tied 1-1 before five minutes elapsed.
Zibanejad opened the scoring at 1:41 when he deflected Patrick Wiercioch's point shot past Ward. The play was reviewed, but the goal was quickly confirmed.
"You've got to look at it as a lucky break on their side, and just get ready for the next shot," Ward said. "The puck was bouncing a little more than I would have liked, but other than that we gave our team a chance to win.
"Unfortunately, it doesn't feel good to lose in a shootout."
Zibanejad improved his career-best numbers to 14 goals and 29 points in 64 games.
"Lately we've been taking it one game at a time. It's cliche but we are," Zibanejad said. "We're doing whatever we can to make sure we get the points.
"The pressure is not on us, the pressure is on the teams above us, and by winning we're putting more pressure on them."
Carolina tied it after Erik Karlsson gave away the puck behind his net, and later was the last player to touch it before it went into the net behind Anderson.
Karlsson tried to sweep the puck around the boards behind the net but fanned on his attempt, giving Skinner possession of the puck behind the goal.
He moved from side to side before banking the puck in off Karlsson to tie it at 4:03.
NOTES: Jason Spezza, Bobby Ryan, Jared Cowen and Colin Greening sat out for the Senators. Mike Komisarek, Radek Dvorak and Justin Peters were scratched by the Hurricanes. ... The Senators recalled Jean-Gabriel Pageau from Binghamton (AHL). ... Senators forward Kyle Turris was chosen as the NHL's first star of last week after posting four goals and three assists in four games. ... Skinner has points in five straight games. ... Nine of the past 11 games between these team have been decided by one goal.