The start of the NBA Europe Live tour marked the Celtics' first game since Garnett and Allen joined the team, creating a "Big 3" with Boston veteran Pierce.
"It was good to finally get out there and get our feet wet," Pierce said. "It was probably the best first game since I've been with the Celtics."
Coach Doc Rivers left both Pierce and Garnett in for 32 minutes, while Allen played 31.
"Doc told us we're not going to play a lot of minutes tonight, but he wanted us to go really hard," Garnett said. "We had a lot of energy tonight."
Bargnani, last year's top overall draft pick by Toronto, played his first game in his hometown in an NBA jersey and scored 13 points in less than 15 minutes.
"Andrea was pretty good. I probably could have done a better job at getting him back in, but I wanted to keep him fresh for tomorrow," Raptors coach Sam Mitchell said. "I know where Andrea's at."
Ford led the Raptors with 15 points.
The Raptors play Roma on Sunday, then travel to Spain to face Real Madrid. The Celtics meet Garnett's former club, the Minnesota Timberwolves, on Wednesday in London.
"We did some good things and we need to work at some other things," Mitchell said. "We play again tomorrow and we'll try to clean some things up."
The sold-out crowd of 11,118 at the Palalottomatica gave Bargnani a standing ovation when he came out for warmups and grew even louder when he scored six of the Raptors' first eight points, including a 3-pointer on his first attempt from the field.
The Celtics finally got going midway through the second quarter when Allen hit a 3 to pull Boston within 40-34. After missing a series of mid-range jumpers, Garnett picked up a rebound off a miss by Pierce and dunked to put the Celtics in front 42-40, their first lead since the opening minutes of the first quarter.
Garnett added a free throw to conclude a 12-0 run for the Celtics and make it 43-40 at halftime.
After his early outburst, Bargnani picked up three fouls and was quiet for the rest of the first half. The Italian started the second half with another flourish, hitting a 3 to restore the Raptors' lead at 45-43, then adding four more points in succession before picking up his fourth foul and returning to the bench.
Garnett was given a technical foul for hanging on the rim after a dunk that tied the game at 52. It appeared he was trying to avoid landing on some players beneath him, and the Celtics argued the call to no avail.
Pierce began hitting his fallaway jumper as the Celtics started to take control midway through the third quarter.
Ford's perimeter shooting for the Raptors kept the game close and Celtics coach Doc Rivers kept Garnett and Allen in for the entire third quarter, which ended with the Celtics leading 70-68.
Reserve guard Gabe Pruitt sealed the game with a free throw for Boston in the final seconds.
Both teams wore special uniforms for the game featuring the colors of Italy's flag. The Raptors' white jerseys had red and green trimming, while the Celtics' lettering was red, white and green, as was the shamrock on their shorts.
Spike Lee, slated to begin shooting a film in Italy next week, sat courtside.
Celtics center Scot Pollard has flown back to the United States for medical exams on his sprained left ankle.