"Our goaltender stole this one for us," Predators coach Barry Trotz said. "We battled pretty hard but we looked like a tired hockey team."
Nashville followed wins at Anaheim and Los Angeles on Wednesday and Thursday by beating the Sharks. The Predators have bracketed a pair of five-game winning streaks around a five-game skid in their past 15 contests.
They got an early goal from David Legwand and the game-winner from Kostitsyn, who has scored in a career-high five straight games.
"We've been streaky," Rinne said. "Hopefully we can turn that around. It's frustrating when you play like that. If we want to be one of the top teams, we have to be more consistent. I think we're headed in the right direction."
Rookie Logan Couture scored the lone goal for San Jose, which has dropped three straight home games this week. The Sharks have lost four straight at home, dating to last month, and have scored just four goals in those losses, including a pair of shutouts.
"I don't know what the answer is," defenseman Dan Boyle said. "The last thing we need on this hockey team is to feel sorry for ourselves. It's very important that we stay together and not go, 'Why us' or 'Why me.' I think we have to find a way."
The most recent blanking by Buffalo on Thursday prompted a scolding by Sharks general manager Doug Wilson, who has grown frustrated by the inconsistent play by last year's Western Conference runner-ups.
While the effort was better this game, the result was the same as the Sharks continue to slip in the standings.
San Jose had a chance to take the lead early in the third period but Rinne made a nice stop on Devin Setoguchi's shot from the side of the net in the opening minute. Rinne also stopped Joe Thornton from in close on a power play a few minutes later.
"We knew they were going to come out and press," Nashville captain Shea Weber said. "We were still a little bit flat-footed. Pekk saved us and gave us a chance to keep battling."
Kostitsyn then provided the tiebreaker. He beat Jason Demers to the rebound of a shot by Patric Hornqvist at the side of the net and knocked it past Antti Niemi to give the Predators a 2-1 lead 6:23 into the third.
"He took a one-timer," Kostitsyn said. "I was just going to the net and the puck was in front of me."
Rinne was tested throughout the rest of the third period, making a particularly tough save against Patrick Marleau a few minutes after Kostitsyn's goal. Rinne then stopped shots from Setoguchi and Thornton on a 3-on-2 with less than eight minutes to go. Couture hit the crossbar in the closing seconds.
"I was getting a lot of chances, they just weren't going in," Couture said. "I think you lose a game and it feels awful. You lose three in a row all at home, in this building, one of the toughest buildings in the league for a team to come in and play, it's just not right. It's embarrassing to lose three in a row."
Couture ended his seven-game run without a goal with the equalizer early in the second period. He skated up ice on a 2-on-1 with Jamal Mayers and showed plenty of patience as he came into the offensive zone before firing a shot that beat Rinne to make it 1-1 with his team leading 19th goal.
One game after being unable to score on 36 shots against Buffalo's Ryan Miller, the Sharks again peppered the net with shots early but were unable to score. Rinne stopped Mayers and Ryane Clowe from in close during one stretch midway through the period and made a glove save after a bad line change against Setoguchi later in the first.
In between those sequences, the Predators took advantage of a turnover in the defensive zone by Boyle, who was unable to control a loose puck with his glove while on his knees. Legwand came up with the puck and fired a shot that Niemi saved with his pad. Legwand beat Setoguchi to the puck and knocked the rebound past Niemi with a backhand to make it 1-0.
- San Jose didn't commit a penalty.
- The Sharks held a moment of silence before the game for former player Tom Cavanagh, who died Thursday at age 28 in what police believe was a suicide.
- Nashville F Nick Saling sat out with an upper body injury sustained Thursday against Los Angeles.