Lindholm tipped Rasmus Andersson's fluttering wrist shot from the point to make it 2-0 at 4:51 of the second.
Matthew Tkachuk took a cross-slot pass from Lindholm and lifted it over Forsberg's outstretched glove on the power play to make it 3-0 at 9:18.
"They worked hard tonight, but I thought we worked smarter," Matthew Tkachuk said. "We had the puck more, we played in their zone more, played solid defensively in front of Dan and he was there to bail us out when we had issues."
Nikita Zadorov's wrist shot from the top of the right face-off circle beat Forsberg blocker side to make it 4-0 at 1:56 of the third.
"Quite frankly we just weren't good enough tonight, just everybody," Ottawa forward Brady Tkachuk said. "We didn't play the game we wanted to tonight, so we have to learn from this."
The Flames are 1-1-1 three games into a seven-game road trip.
"I think physically we were really, really engaged," Sutter said. "Those [previous] two games in Montreal and Toronto are hard games. They take a lot out of your team. Besides the travel, they're hard games to play. So this is a really good gut check for our team. They did a really good job. Mentally, they're probably not as sharp as they need to be, but physically they were really good."
NOTES: The Flames' six road wins are tied with the Minnesota Wild and New York Rangers for the most in the NHL. … Batherson, who leads Ottawa with 16 points (seven goals, nine assists), entered COVID-19 protocol a couple hours before puck drop. … Forward Andrew Agozzino had one shot in in 6:52 in his Senators debut.