Eller and Max Pacioretty scored for Montreal in the shootout; Peter Holland scored for Toronto on the first attempt.
"You have to like it," Eller said. "It's what you live for as an athlete. Of course I was a bit nervous, but you have to be a little bit nervous when you are put in that situation. It's fun. It's where you want to be."
Toronto (17-20-9) has lost two in a row. The Maple Leafs were outshot 14-3 in the first period.
"We gave the players the day off (Friday)," Toronto coach Mike Babcock said. "We won't be doing that again."
Mike Condon made 17 saves and stopped four of five shootout attempts for Montreal (24-20-4).
The Canadiens, who had eight goals in their prior five games, took a 1-0 lead when David Desharnais scored at 1:41 of the first period. He took a pass at the side of the net, and with Maple Leafs goalie James Reimer caught flat-footed, was able to snap a quick shot high stick-side.
"We had a lot of trouble in the first period of games the past few weeks, so getting that first goal and then getting another one gave us a lot of confidence in the third period," Condon said. "They came out and kind of took it to us in the second period and started chipping away at our lead, but we stuck with it and I am very happy with the outcome."
Toronto did not get its first shot on goal until 11:35 into the game; by then, Montreal had 12.
Montreal made it 2-0 at 18:33 when Tomas Fleischmann snapped a shot from the slot.
The Canadiens thought they took a 3-0 lead at 7:43 of the second period when Daniel Carr, called up from St. John's of the American Hockey League, took a shot that Reimer caught in his glove. It looked like he may have put it into the net, but the NHL Situation Room reviewed the play and ruled the puck did not cross the goal line.
Toronto center Nazem Kadri made it 2-1 at 8:14 when he accepted a pinpoint backhand pass from defenseman Morgan Rielly in the slot.
The Maple Leafs tied it 2-2 at 4:37 of the third period when Condon stopped Holland's wrist shot only to have Joffrey Lupul tap home the rebound from the top of the crease. It was Lupul's 11th goal of the season, second in the past three games, after going 11 games without scoring.
Reimer kept it tied when he stopped Brendan Gallagher on a breakaway at 7:39. Gallagher deked to his backhand, but Reimer stayed with him all the way.
Gallagher later got the puck off a faceoff in the Toronto zone, turned and took a shot that hit the post and crawled along the goal line.
In the shootout, the Canadiens' first two shooters, Alex Galchenyuk and Brian Flynn, each took a shot that Reimer bobbled and dropped on the goal line. Eller said he wondered if Montreal's bad luck was about to continue.
"I can't tell you how often in the last while we have been on the wrong side of those things that make the difference in the outcome of the game," Eller said. "Finally we were on the right side of that today and I think we deserve it."
