Crosby had assists on all three goals for Pittsburgh, which extended its winning streak in the series to eight games.
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The Penguins also halted the Lightning's bid to tie a club record with an eighth consecutive triumph on home ice.
Kunitz opened the scoring seven minutes in, taking a pass from Crosby and lifting it past Tampa Bay goalie Ben Bishop from the left circle. Kunitz sealed the win, scoring into an empty net with 1:43 remaining in the game.
Crosby's accomplishment caught the attention of his coach.
"It is amazing how often and how fast he has been on that score sheet," Penguins coach Dan Bylsma said of Crosby, whose second assist was the 700th point of his career.
Crosby, the NHL leader this season with 36 points, appreciates the milestone.
"You obviously play to win, but when you get things like that, little individual things, I guess you enjoy it for a little bit," he said. "It's fun to see that, but ultimately you play to win games. It's nice we were able to get a win in the same night."
Brandon Sutter scored the second Pittsburgh goal on a power play midway into the third period. Sutter took his own rebound behind the net and banked it off the leg of Lightning defenseman Victor Hedman on the opposite side.
"It's part of the game. It was a really lucky bounce for them and unfortunate for us," Hedman said. "Down one goal and they get a lucky one on the power play, so obviously it sucks a little momentum out of you. ... It could have gone either way. They got the bounces today."
Tampa Bay had several excellent scoring chances. Richard Panik got loose in front of the Pittsburgh net just before the second intermission but was stopped by Fleury, who earlier made a save on Nikita Kucherov on a partial breakaway.
"Part of the problem tonight was we let Fleury see shots, not all of them, but too many," Lightning coach Jon Cooper said. "And we were really having a tough time getting pucks to the net."
The Lightning also came up empty in three man-advantage situations.
Bishop, who finished with 26 saves, has lost twice in his past three games.
The early goal set the tone for Pittsburgh, which has won all but one of the 15 games in which it scored first. The Penguins are 9-0 when leading after the first period, and Tampa Bay is 0-6 when behind after one.
Crosby, Kunitz and C Evgeni Malkin each have seven-game point streaks against the Lightning, with a combined 23 goals and 22 assists. ... Pittsburgh broke a tie with Tampa Bay for second place in the Eastern Conference. ... Penguins LW James Neal was held without a point for the first time in six games, having collected 11 on six goals and five assists in his previous five outings.
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TAMPA, FL - NOVEMBER 29: Ondrej Palat #18 and Tyler Johnson #9 of the Tampa Bay Lightning talk during the pre game...