Nashville (44-21-8), which won at home for the first time in its past six games, tied the St. Louis Blues for first place in the Central Division.
"The last stretch here, we've lost some games in the third period," Rinne said. "It's been a little bit of a topic that we've been talking about, so for sure it's nice to finish strong and not give up a goal or anything like that. Just do it the right way."
Buffalo (20-44-7) is 1-7-2 in its past 10 games. Matt Hackett made 40 saves.
"They're a very good team," Sabres coach Ted Nolan said. "For us to keep battling and fight through and do what we did, especially in that third period, we had a number of scoring chances. We hit the goal post twice and [Tyler Ennis] had a couple of breakaways, so I can't fault these guys for the effort for sure."
The Predators had three consecutive days without playing a game for the first time since the All-Star break.
"There's no question that the rest helped," Nashville coach Peter Laviolette said. "The schedule is the schedule and we're not going to complain about it. There's 27 games, I think there was, in 49 days, whatever it was, and we went 13-11-3. It's not a great record.
"There was just no opportunity to get work in in practice and really probably not get enough rest either, but everybody's dealt with that. Everybody's got to play 82 games somehow some way, no matter whether it's tough at the beginning or tough at the end, tough in the middle, it doesn't matter. Everybody's got the same hand dealt to them."
Taylor Beck gave the Predators a 1-0 lead at 14:52 of the second period on a power-play goal. Defenseman Shea Weber took the original shot from the point and Beck redirected it past Hackett for his seventh goal of the season. The Predators' power play was 4-for-46 in the past 15 games before Beck's goal.
"We've been struggling a little bit lately, and we wanted to get a lot of pucks to the net," Beck said. "Fortunate enough, (Weber) got a good shot off, and I got a tip on it. That stuff happens every once in a while."
Mike Fisher gave the Predators a 3-0 lead at 15:01 of the third period on a wrist shot from the slot. Forward Viktor Stalberg made the pass from behind the net and Fisher took a shot that beat Hackett on the stick side for his 17th goal of the season.
"I thought that we didn't really change anything," Rinne said. "We just tried to keep it going, tried to score the next goal. I like that aspect of our game. We always want to score that next goal, and that's what happened in the third and it obviously gave us more breathing room. I'm happy with how we finished the game. We didn't sit back or anything like that."
Predators forward Matt Cullen had two assists. He has four assists in the past four games.
"It's been a while since we won here," Ellis said. "To come out with that performance and Pekka back to how he was playing earlier in the year, it was huge for us. We've just got to keep going. The next one is Tuesday (against the Montreal Canadiens), and it's just as important as this one."