Staal took a pass from Evgeni Malkin, stickhandled into the heart of the Toronto zone and fired a wrist shot past goaltender Andrew Raycroft for his 23rd goal of the season to end it.
Also scoring for the Penguins were Malkin, Ryan Malone and Maxime Talbot. Malkin finished with three assists, and Sidney Crosby earned one assist to increase his NHL-best points total to 87.
Nik Antropov, Mats Sundin, Bates Battaglia, Bryan McCabe and Boyd Devereaux scored for the Leafs in front of a regular-season record Air Canada Centre crowd of 19,620.
Pittsburgh remained tied for fourth in the East with Ottawa with 67 points each. The Penguins are on a 11-0-2 run with 24 of a possible 26 points in their last 13 games. They haven't lost in regulation time since a Jan. 10 road game against the Florida Panthers.
The Leafs moved within one point of the idle Tampa Bay Lightning, who are holding the eighth and last berth.
Toronto wiped out a three-goal deficit with three consecutive goals in the closing seven minutes of the second period.
McCabe put the Leafs up 4-3 at 1:05 of the third when he slammed in a long shot for his 10th goal of the season with goalie Marc-Andre Fleury helpless behind a screen of legs.
Malone tied it 4-4 with his ninth off a rebound at 1:59.
Devereaux put Toronto up 5-4 off a rebound at 11:44. It was his third goal in 15 games since being summoned from the AHL Toronto Marlies.
Talbot tied it 5-5 with his seventh at 13:49 with a high shot from the circle to the right of Raycroft.
With just over six minutes left, Leafs forward Kris Newbury was taken off on a stretcher. Newbury got into a fight with Ronald Petrovicky and appeared to be out on his feet after Petrovicky landed a punch. Newbury struck his head on the ice and a pool of blood was scraped away as he was wheeled away.
Staal had opened the scoring at 19:22 of the first period. Evgeni Malkin fed Michel Ouellet who set up Staal in the circle to the right of the net. The passing sequence happened so fast that Raycroft couldn't get over in time to stop Staal from scoring his 21st goal.
Staal scored again at 3:32 of the second period, with Malkin getting another assist, when he beat Raycroft from the circle to the goalie's right with a shot that planted the puck in the net's far top corner.
The 18-year-old native of Thunder Bay, Ont., got his 22nd goal on his 80th shot, improving his shooting percentage to a league-best 27 per cent.
Malkin got his team-high 27th goal - and his 13th on power plays - at 8:44. Crosby worked some magic at the blue-line when he sucked Hal Gill towards him and sent a pass behind his back to spring Malkin on a 2-on-1 rush with Mark Recchi. The 20-year-old Russian used Recchi as a decoy and fired in a low wrist shot for his third point of the game. He leads all NHL rookies with 62.
With 30 minutes gone, the Leafs were outshooting the Penguins 14-11 but were down 3-0. That would quickly change.
Antropov started Toronto's rally by barging onto Fleury's doorstep to convert a pass from behind the net from Alexei Ponikarovsky at 13:25. It was the ninth goal for the six-foot-six Kazakh, and Sundin earned his 700th career assist on the play.
Sundin added a log to the fire with his 22nd at 16:44. Fleury got a piece of an Antropov shot, the puck dropped behind him, and Sundin was there to tuck it into the open net.
Battaglia tied it 3-3 with his ninth at 19:05. Chad Kilger had the puck behind the net when he spotted Battaglia rushing into the circle to the left of Fleury. Battaglia buried it in the far side of the net.
Notes: Toronto had a 28-25 shots edge through regulation time, and Pittsburgh got all three shots recorded in overtime . . . On power plays, Toronto was 0-for-3 and Pittsburgh 1-for-3 . . . The average age of the Penguins is 27.1 years. San Jose (26.8) is the only younger team in the NHL . . . Pittsburgh has three players 20 or younger with at least 20 goals - Malkin (27), Crosby (25) and Staal (22) - and that hasn't happened since 1981-82 when Edmonton had Wayne Gretzky, Mark Messier and Paul Coffey . . . Wade Belak hasn't scored a goal since Dec. 20, 2003 . . . Toronto's next game is Tuesday when the New York Islanders visit. The Leafs play in Philadelphia on Thursday.