"I think one of the best things we did was blocking out, blocking shots, clearing rebounds -- we really didn't do much wrong tonight defensively," said Ellis, who was making only his second start of the season in place of Cam Ward.
"We played very, very well. We limited our turnovers, when we did have any turnovers, we had great back pressure, we had great angles. You look at our whole defensive game tonight -- it was phenomenal."
It was Ellis' 13th career shutout and first since Nov. 9, 2010, against Toronto when he was with Tampa Bay.
Ellis has only allowed a total of one goal in his two starts for the Hurricanes this season, while Ward has yet to give up less than three goals in one of his starts.
With the win, Carolina improved to 9-0-1 in its last 10 home games against the Senators.
Ottawa came in with a 5-1-1 record but struggled to generate many scoring chances. Jason Spezza underwent surgery on a herniated disc in his back earlier Friday, and the Senators could be without their No. 1 center for up to two months.
"I thought Ellis was good for them, (but) I thought we were on the outside a little bit too much," Ottawa coach Paul MacLean said. "Our energy level wasn't as good."
Craig Anderson made 34 saves for Ottawa, which was trying to win its third game in four nights. Anderson has allowed only seven goals in seven games and entered the game leading the NHL in save percentage.
Staal took advantage of a strong effort by Alexander Semin to score his goal.
Skating along the boards, Semin maneuvered around several Ottawa players before cutting toward the center of the ice. He then shot the puck into traffic in front of the net, and Staal managed to tip it over Anderson's shoulder.
"Alex gave us a great game," Carolina coach Kirk Muller said. "He's a player that's been in a lot of big games, he's really been committed and made some great plays and (been) very valuable for our hockey club so far."
The Hurricanes signed Semin to a one-year contract in the offseason with the idea of pairing him with Staal.
"He's a guy that just backs off defenses," Staal said. "If they're going to be aggressive, he's got the skills to make them pay with a one-on-one move that not a lot of guys can do."
Carolina's Jeff Skinner almost scored midway through the second when he hit the crossbar from just outside the crease.
The Hurricanes' power play continued to struggle, squandering all four chances. Carolina is 3-for-30 on the season.
The Senators had been 4-1-1 this season when allowing opponents to score first.
Ottawa had a four-minute power play in the second period when Carolina defenseman Jamie McBain cut Chris Neil with a high-stick, but the Hurricanes' defense didn't allow the Senators to threaten Ellis.
"I don't think it was our best game, but over the course of the schedule, you're not always going to play your best game," MacLean said.