The teams were tied 2-2 after two periods and didn't score in the third or overtime.
Spezza, who scored the lone goal in Ottawa's win Saturday against Toronto, stopped at the circle during his shootout attempt, stickhandled around goalie Jeff Zatkoff, and scored. Kyle Turris added a goal in the shootout for the Senators.
Turris scored his 26th goal in regulation, and Mark Stone had his fourth for Ottawa, which will miss the playoffs for the second time in six years and first since 2011.
Jokinen scored on the power play in regulation, his 21st of the season, and Lee Stempniak added his 12th for the Penguins, who rested captain Sidney Crosby and several others for their final game before the playoffs.
Pittsburgh will face Columbus in the first round of the playoffs.
It will be the first postseason meeting between the teams, separated by a four-hour drive. The perennial-contending Penguins have lots of star power and three Stanley Cup titles. Columbus, during its first season in the Eastern Conference, secured its second playoff berth and first since the 2009 season.
Despite missing the finale, Crosby won his second scoring title - topping Anaheim's Ryan Getzlaf.
Crosby, who finished with 36 goals and 104 points, outscored Getzlaf by 17 points to win the Art Ross Trophy for the first time since the 2006-07 season, his second in the NHL.
The seven-year gap is the second-longest span between scoring titles.
Pittsburgh hopes to have former NHL MVP and two-time scoring champion Evgeni Malkin back for the playoffs after he missed the final 11 games of the regular season because of a foot injury.
Jokinen opened the scoring Sunday with a power-play goal with 5:14 left in the first period. His centering pass deflected off Eric Gryba's skate and trickled between Lehner's pads.
Turris tied it later in the period, scoring in the final seconds of a power play when he sent a shot past Zatkoff from the left faceoff dot.
Stempniak put the Penguins ahead 2-1 when he tipped Bennett's shot behind Lehner at 5:16 of the second.
Stone tied it with Ottawa's second power-play goal at 17:14 from the top of the crease, whacking in a rebound of Patrick Wiercioch's point shot.
NOTES: The Penguins played their fourth straight overtime game. ... A Penguins player has won the scoring title in 15 of the last 26 seasons. ... Crosby and Malkin are the fourth set of teammates to win multiple scoring titles while playing together and the first since Boston's Phil Esposito and Bobby Orr between 1969-75. ... Ottawa scratched Jared Cowen, Matt Kassian and Mika Zibanejad.