Jason Arnott scored the tiebreaking goal 5:02 into the third period, lifting Nashville to a 4-3 victory over Edmonton on Tuesday.
Grebeshkov had a goal and an assist while making his Predators debut against his former team, one day after being acquired from the Oilers in exchange for a second-round pick in this June's NHL Draft. Colin Wilson and Shea Weber also scored for the Predators.
On the game-winner, Arnott threw the puck in front of the Edmonton goal from behind the net and it bounced off the right skate of Oilers defenseman Jason Strudwick into the net.
"We've seen those a lot this year, we're not handling the front of the net," Edmonton coach Pat Quinn said. "We don't get those kinds of goals at the other end. We're too busy with the drop passes and the lateral plays, all that pretty stuff that doesn't end up in the net."
It was the Nashville captain's 15th goal of the season.
"We came out strong, but a little rusty," Arnott said. "The first period, we were getting our legs under us just getting back into the game skating in a game. You can practice all you want, but you need to get out there and play in games."
Mike Comrie, Gilbert Brule, and Marc Pouliot scored for Edmonton.
Neither team led by more than a goal, and Pouliot tied the score 3-3 at 2:54 of the third before Arnott put the Predators ahead for good.
Brule tied the game at 2 at 11:53 of the second when he deflected Theo Peckham's wrist shot from the left point past Predators goalie Pekka Rinne on the stick side.
Nashville regained the lead at 15:12 on Weber's power-play goal off of an assist from Ryan Suter.
Two days earlier, the Predators' top defensive pairing had squared off in the Olympic gold medal game in Vancouver. Suter, who won silver with the United States, sent a pass from the top of the right circle to Weber in the center of the blue line where the gold medal-winning Canadian's one-timer beat Jeff Deslauriers on the glove side.
"It was fun to be back playing with him," Suter said. "Hopefully we can continue to have fun and continue to do these things."
Comrie scored the game's first goal at 6:37 of the opening period. Tom Gilbert carried the puck into the Nashville zone with Comrie and Shawn Horcoff on a 3-on-1 against Suter. From the right side, Gilbert sent a pass to Horcoff, whose shot was stopped by Rinne. However, Comrie was parked just to the left of Rinne and had an easy tap-in for his seventh goal of the season.
Wilson tied it for the Predators at 12:14 of the first on a wrist shot from the high slot that Deslauriers never saw thanks to a screen by Jordin Tootoo. Grebeshkov created an Edmonton turnover inside the blue line and was credited with an assist on Wilson's goal.
"The big thing on the first goal was that Toots was in front," Nashville coach Barry Trotz said. "I liked that line tonight."
Grebeshkov scored his goal at 14:55 of the first when his slap shot from between the circles beat Deslauriers on the glove side.
"I was waiting for that line to open up, and as soon as it did, I took my chance," Grebeshkov said. "I'm really glad that our team won tonight and that I helped them."
The Predators played their first home game since their building's name was changed to Bridgestone Arena late last month. ... Nashville has scored at least one power-play goal in a season-high seven consecutive games. ... The Predators are 19-1-0 when leading after the first period this season. ... Peckham's assist was his first career NHL point.