Bobby Ryan also scored a shootout goal, and Mike Condon made 31 saves for the Senators (25-15-4), who swept a three-game road trip.
"Huge. We knew coming in today that it was going to be a battle for us," Ottawa captain Erik Karlsson said. "We were already battling last game [at the Columbus Blue Jackets, a 2-0 win Thursday]. We knew that we were going to be a little bit mentally and physically tired, which is normal at this time of the year. These are the games that you need something to fall back on, and I think we executed well."
Toronto (21-14-9) is 1-6 in the shootout this season. Auston Matthews, James van Riemsdyk and Tyler Bozak each failed in his attempt; Mitchell Marner scored a goal that was upheld after review.
"I don't know, maybe other goalies are good, maybe we're not that good at it," Babcock said. "We work on it every single pregame skate. It's something we'll have to get better at because we've left points."
Ryan scored his 10th goal of the season when he deflected a point shot from Marc Methot at the top of the crease at 14:02 of the first period to put Ottawa up 1-0.
Toronto outshot Ottawa 15-4 in the second period, and Bozak scored his 12th goal of the season on the power play at 7:31 to tie it 1-1. Matt Martin scored his fourth goal of the season 7:00 into the third period to give Toronto a 2-1 lead.