Daniel Rubin scored in the third period to give the Swiss a 1-0 lead before Eberle tied it late.
Devan Dubnyk, also of the Oilers, was solid in goal for the Canadians.
Canada coach Brent Sutter said he was happy with his team's performance but added there's still a lot of work to do.
"I thought especially in the first two periods it was a really high-paced game. Both teams had some quality scoring chances, both goalies played well," Sutter said. "For our first game and only having been together for 48 hours as a group and having only had two practices I thought we handled it all pretty well. To get a win is a confidence boost."
Sutter coached the Calgary Flames for the last three seasons and is seeing Eberle, Dubnyk and fellow Oilers teammate Ryan Nugent-Hopkins from a different perspective behind Canada's bench.
"No question Eberle played really well. I thought both him and (Nugent-Hopkins) were both really good and you can tell they've obviously played together," Sutter said. "They make things happen but they're also dependable players defensively too and it's always nice to have that from your top players.
"I thought Dubnyk made some big saves at key times and overall as a group I was happy with the work ethic and the competitiveness. Now it's just about working on our details."
Sutter acknowledged that any player or coach at the world hockey championship representing Canada has had a disappointing end to their NHL season.
"Any player who doesn't make the playoffs or gets beat out in the first round, you're always disappointed," Sutter said. "But then you get the opportunity to represent your country in a tournament like this — it's like a second chance."
The same holds true for Sutter, who parted ways with the Flames at the end of the regular season after missing the playoffs for the third straight campaign.
Despite not having a job in the NHL, he doesn't see this tournament as an audition for future employment.
"For myself I don't look at it as though I've got anything to prove. I'm here to coach this team and trying to succeed and I was selected by Hockey Canada to be the head coach and very proud and honoured to do that," Sutter said. "I don't get caught up in the fact that this is something that's going to make or break my career with coaching. I know that I'm unemployed in the National Hockey League and that's just the way it is.
"That's coaching and that's the way it goes when you don't have success. That happens and you have to hold your head up and move on."
Canada and Switzerland will meet in another exhibition game Tuesday in Kloten, Switzerland.
Canada opens the tournament May 4 in Helsinki, Finland, against Slovakia.