Streit's backhand shot from the right circle caromed between Fleury's pads and into the net as the Flyers assured themselves of a matchup with the New York Rangers when the Stanley Cup playoffs begin next week.
Jakub Voracek scored twice for the Flyers, and Claude Giroux added his team-high 28th goal of the season. Ray Emery made 12 saves after replacing injured starter Steve Mason to start the third period.
Jussi Jokinen, James Neal and Kris Letang scored for the Penguins. Fleury stopped 17 shots and hardly looked sharp while giving up the game-winner. Letang scored in the final minute of regulation to force overtime after Giroux gave the Flyers a 3-2 lead with 1:15 left.
Pittsburgh will play either Detroit or Columbus in the opening round of the postseason.
Philadelphia's victory means the rivals will have to wait at least one series before facing each other in the playoffs, if they see each other at all.
That might be a blessing for the Penguins, who lost the season series to Philadelphia 4-1 and were shoved out of the postseason by the Flyers two years ago.
Philadelphia did its best to avoid picking up the two points needed to set up a showdown with the Rangers. The Flyers gave away a one-goal lead twice in the final 5 minutes of regulation.
Neal made it 2-2 when he beat Emery with 5:07 remaining. Emery took over in goal for Mason, who made 21 saves but sustained an upper-body injury when he was bowled over in the crease late in the second period.
Giroux appeared to give the Flyers the lead for good when he deflected in a pass from Voracek with 1:15 to go. Pittsburgh pulled Fleury, and Letang tied it with 36 seconds left, scoring his first goal since his return from a stroke suffered in January.
The teams began the extra period with a rare 3-on-3 following a dust-up in front of the benches at the end of regulation. Both sides were returning to full strength when Streit raced down the right side and flipped an innocent looking backhand at Fleury that the goalie misplayed.
Jokinen gave Pittsburgh the lead 2 minutes into the game when he converted a pretty drop pass from Neal into a knuckling wrist shot over Mason's shoulder for his 20th goal of the season, the first time Jokinen has reached that milestone since scoring a career-high 30 for Carolina in 2010.
Voracek tied it with a power-play goal with 2:51 left in the first, beating Fleury low. It was the last Flyers shot for nearly 17 minutes.
The Penguins spent most of the second period tilting the ice but couldn't find a way to beat Mason. The NHL's top-ranked, power-play unit couldn't convert during a 5-on-3 advantage that lasted more than 90 seconds. A one-timer by Neal clanged off the right post, and Mason made a lunging glove save from his belly to deny Beau Bennett.
NOTES: The game was the 400th for Penguins coach Dan Bylsma, who is 252-117-31 since taking over midway through the 2008-09 season. Only Eddie Johnston (516) has coached more games in team history. ... The crowd of 18,673 tied the largest at Consol Energy Center since it opened on Oct. 7, 2010. ... Philadelphia D Hal Gill played 15 minutes in his first game since Dec. 21. Gill had been a healthy scratch in 44 straight games. ... Both teams will end the regular season on Sunday. Pittsburgh will host Ottawa, and the Flyers will face Carolina in Philadelphia.