Young had eight points in the third, including a pair of 3-pointers, and finished 7-for-15 from the field.
"He kind of got us back in the game in the third quarter, making those 3s," 76ers coach Maurice Cheeks said.
Cheeks said Speights helped win it with his energy in the fourth.
"He played well," Cheeks said. "He stepped in and made some shots and defended well."
Elton Brand had 10 points and nine rebounds in 29 minutes for Philadelphia.
"He's settled himself in once we've been out on the court," Cheeks said. "This is our third game, so it gives him a little more confidence playing with his new team. Practice is always a little different than a game."
Brand said he's feeling comfortable with the 76ers, who signed him team during the summer as a free agent.
"It's early, but I really love this team," Brand said. "Our second unit played really well."
Toronto's Chris Bosh was held to just four points in 23 minutes, going 1-for-5 from the floor. Jermaine O'Neal had nine points in 25 minutes for the Raptors, who lost for the first time in 12 preseason games at Air Canada Centre.
Solomon fouled Williams with 0.1 seconds remaining in the third on a 3-point attempt and Williams made two of three to tie it at 60.
Kapono tied it at 77 on a jumper with 3:48 to play, but the 76ers scored eight of the final 10 points, including two dunks by Speights and a game-sealing dunk from Igoudala with 12.9 seconds remaining.
Philadelphia will host Toronto in the regular-season opener for both teams on Oct. 29.
"This is the team we open up with so we wanted to go out and learn as much as possible," said Raptors forward Jamario Moon, who had seven points and 10 rebounds.
Toronto led 41-39 at the half.
Toronto was without F Joey Graham (hamstring), C Nathan Jawai (heart) and C Jamal Sampson (wrist). Philadelphia did not have forward Jason Smith (knee). Former Raptor Donyell Marshall also didn't play for the 76ers, and G Roko Ukic did not play for Toronto. ... Philadelphia outrebounded Toronto 51-37. Toronto opponents have posted back-to-back 50 rebound games. ... Actor Ving Rhames attended the game.