The Belgians started as the better team but Italy withstood the attack and regained their footing after the opening 20 minutes.
The Azzurri were rewarded just after half-hour mark when Leonardo Bonucci sent in a pinpoint cross from the midfield line over the Belgium defence, and Giaccherini took one touch to settle it before shooting past Thibaut Courtois for the opener.
Italy nearly doubled their advantage five minutes later when Belgium failed to clear their lines following a corner, but Pelle headed the ball just wide of the post from six yards.
Belgium pushed back before the break as Marouane Fellaini played in Kevin De Bruyne, but Giaccherini tracked back to get his foot to the Manchester City man's shot.
After half-time, Belgium nearly equalised on the counter-attack as Romelu Lukaku's first-time shot curled inches past the post.
There were clear chances for both sides as the match drew to a close but it was Antonio Conte's men who had the last word, with Antonio Candreva crossing for Pelle to smash home following a counter attack.
Gerard Pique scored a late goal as defending champions Spain began their campaign with a 1-0 win against a stubborn Czech Republic side.
The Group D match had seemed set to end in a goalless draw despite plenty of Spain pressure, but Pique sealed the points when he headed home a fine Andres Iniesta cross three minutes from time.
Czech Republic keeper Petr Cech was called upon to make a succession of saves during the match, although David De Gea in the Spain goal also had to be alert.
Spain began to apply heavy pressure as the game drew towards a close and got their reward when Iniesta set up his Barcelona teammate for a deserved winner.
Ireland were far brighter during the first half, and they came close to the opening goal when Derby midfielder Jeff Hendrick curled a superb shot against the crossbar.
The second half began at a thrilling pace, and Ireland -- having just seen Hendrick go close again -- took the lead through Wes Hoolahan's neat finish on 48 minutes, firing in a curling shot on the bounce from Seamus Coleman's cross.
Sweden improved after the goal, and they drew level on 71 minutes when Zlatan Ibrahimovic latched onto John Guidetti's neat flick and then sent in a cross that Clark headed past Darren Randolph at the near post.