Roman Josi scored his 150th NHL goal, and Forsberg had a goal and an assist for Nashville (19-14-6), which went 5-for-5 on the penalty kill.
"I thought the penalty killers did a really good job tonight," Predators coach John Hynes said. "Obviously, we got some big saves from [Saros] on the penalty kill, but their power play is threatening. They had a lot of different looks. They attack you at the top, they do a really good job down around the goal line, obviously with [Brady] Tkachuk. That was a big difference in the game. One of the goals was to stay out of the box, and we certainly didn't do that as much, but the penalty kill came through."
The Predators are 5-0-1 in their past six games.
"Even before we started winning, we were playing better hockey," Forsberg said. "And I thought there was a stretch where we didn't get the results, but we were playing better and building our game. And now, obviously the results have been coming. But at the same time, we've got more than half the year left. We've got to keep building our game and keep finding wins."
Cam Talbot made 33 saves for Ottawa (18-19-3), which was shut out for the first time this season.
"Our power play has won us plenty of games," Senators coach D.J. Smith said. "Tonight was a night where we needed them to score, and they didn't, but I have no doubt that they'll get hot."
Josi gave Nashville a 1-0 lead at 14:29 of the first period when he one-timed a pass from Nino Niederreiter on a 2-on-1.
"It's always nicer if you get a win," Josi said of scoring his 150th. "It's pretty cool. I wasn't even aware of it. [Ryan] McDonagh actually told me. It's pretty cool. I've been fortunate to play in this league for a long time, and obviously I'm enjoying the moment, so it's definitely pretty cool."