The Italy international's 32nd-minute strike and a second-half header from substitute Valentin Roberge handed interim boss Kevin Ball a win at the first attempt and just a second of the season for the Black Cats in all competitions.
However, while Sunderland face much more difficult tests in the weeks ahead - Liverpool are due at the Stadium of Light on Sunday and Manchester United will follow them both intent of keeping the Wearsiders marooned at the foot of the Barclays Premier League table - it will be the manner of the performance which will have most pleased Ball after a traumatic few days for the club.
The temporary manager made something of a statement before kick-off when he named only three of Di Canio's summer signings - Giaccherini, Ki Sung-yueng, and Jozy Altidore - in his starting line-up and restored former skipper Lee Cattermole, Carlos Cuellar and Sebastian Larsson to the team.
But more importantly, he had picked up the pieces after a chaotic weekend and knitted together a side from the wreckage of the Italian's ill-fated reign.
Giaccherini, who had been hauled off at half-time at West Brom on Saturday, and Cattermole, who was left in no doubt that he was surplus to requirements during the summer, proved particularly influential as the home side dominated the first half, and the goal when it arrived was long overdue.
Cattermole's swinging cross just needed a touch in front of goal and Giaccherini duly obliged.
It might have been 2-0 before the break, but Altidore was unfortunate to see a well-struck effort cannon back off the post.
The Black Cats needed a second goal to kill of Darren Ferguson's game side, but they had to wait until the 73rd-minute when Roberge met Adam Johnson's cross with a firm header to effectively end the contest, to the delight of the home fans among a crowd of 18,126.
Early League One promotion contenders Posh had scored a total of 11 goals in the previous two rounds, but rarely looked like causing keeper Keiren Westwood any real problems.
Where disorganisation and fear of failure had crippled Di Canio's team, Ball's men played with confidence and at times, even a swagger, albeit against more modest opposition than they faced in all but one of the Italian's 13 games at the helm.
The former Sunderland midfielder opted to field Giaccherini in a free role behind lone striker Altidore, and with Cattermole providing cover in front of the back four - the former Swindon boss had spent much of the summer pleading in vain for such a player to be recruited - there was a much more compact look about the Black Cats.
Larsson fired wide from distance with six minutes gone and Posh keeper Bobby Olejnik had to block Altidore's first-time shot after Giaccherini had played him in.
The Italy international was a central figure throughout the first half, firing wide from a tight angle with 25 minutes gone and then failing to make the most of Jack Colback's cross in front of goal seconds later.
However, it was he who eventually made the breakthrough 13 minutes before half-time when he timed his run to perfection to meet Cattermole's inviting cross and half-volley home from close range.
Giaccherini was mobbed by all his outfield team-mates in a show of unity which said much about the prevailing atmosphere in the dressing room.
Altidore rattled a post with a 44th-minute drive as Sunderland negotiated the first half in relative comfort with Danny Swanson testing Westwood from distance after four minutes before both he and substitute Joe Newell were less accurate from similar range.
Peterborough returned determined to force their way back into the game, but Giaccherini saw a shot blocked by Jack Payne and Johnson curled a left-foot effort just wide with Olejnik at full-stretch.
However, it was Westwood who was called into action with 22 minutes remaining when he needed two attempts to claim striker Britt Assombalonga's skidding shot.
But victory was secured 17 minutes from time when, after a left-wing corner had been cleared, Johnson crossed for Roberge to head home.