This is the season of mellow mists and fruitfulness and, perhaps appropriately, Sunderland experienced both on Tuesday night. As fog shrouded the stadium the mood within it seemed emollient as the club began life after Paolo Di Canio.

The calm following the storm whipped up by the former manager produced a safe passage into the fourth round under Kevin Ball's assured, temporary stewardship.

Hardly surprisingly given the recent amalgam of appalling results and off-field turbulence, little more than 18,000 fans turned out, filling Sunderland's home to less than half of its 49,000 capacity.

Those that made the trip will have deemed it worthwhile after Emanuele Giaccherini's half volley, Valentin Roberge's header, Lee Cattermole's midfield bite and Jozy Altidore's powerful penalty-area nuisance value saw off Darren Ferguson's Peterborough United in fairly comfortable fashion. "Don't worry, be happy," they sang. Goodness knows what Di Canio would have made of the laid-back reggae vibe.

Sunderland's self styled "unique one" had rarely stopped talking about his much-vaunted Wearside revolution but Ball, the development coach turned caretaker manager, suggested he is more into evolution. Or should that be counter-revolution.

Only three of Sunderland's 14 summer signings – Altidore, Giaccherini and Ki Sung-yueng – were in an XI featuring a return to action for the recently overlooked Carlos Cuéllar and a first start of the season for Cattermole. Indeed as the latter spiced up central midfield it seemed the clock had been rewound to the Martin O'Neill or possibly Steve Bruce era.

Out went the Italian's bold 4–2–4 formation to be replaced by Ball's comparatively conservative 4–2–3–1 configuration involving Giaccherini's deployment in the centre of the attacking trio behind Altidore.

Considering a shoulder injury seems set to sideline Steven Fletcher, Sunderland's most prolific striker, for several weeks, Ball's options were limited. No matter, although Giaccherini's left foot could not quite connect with a defence bisecting cross from Jack Colback - very good at left back - they did not take too long to pay dividends.

It was the 32nd minute when Cattermole's well-flighted cross into the box fell to the advancing Giaccherini who, at full stretch, half-volleyed Sunderland ahead, beating Bobby Olejnik with his left foot from close range. As the Italy international, signed from Juventus, celebrated with a knee slide, a smile spread across Ball's face. Within seconds every home outfield player had joined in the celebrations. "Lee, Lee, Cattermole," chorused the stadium in homage to Di Canio's nemesis.

In a single swipe of a left boot the fog which had rolled in off the North Sea seemed to lift momentarily. "I thought it was a really good performance," said Ball. "We worked very hard and created quite a lot of chances. We created a decent tempo and passed quickly. The players put in a really good shift and Lee Cattermole was fantastic, outstanding. He drove everyone else on. He was excellent."

Not that Ball took Ferguson's side lightly. As befits a team riding high in League One, Peterborough were proving obdurate opponents, missing half-chances when shots from Jack Payne and the substitute Joe Newell flew off target.

Although Sunderland would have gone into half-time two goals up had Altidore's shot not bounced off the base of a post, the second half saw Keiren Westwood finally called to arms to make saves from Newell and Britt Assombalonga. Even so, the sight of Ferguson looking as angry as his father sometimes used to in Manchester United's technical area confirmed that Ball's team were on top.

Peterborough like to control possession but, pressing them high up the pitch, Cattermole and company were not about to let the visitors have it. Sunderland duly doubled their advantage when Adam Johnson – one player apparently reborn under Di Canio's strict regime – crossed superbly from the right and Roberge, who minutes earlier had replaced the injured John O'Shea, thumped a header beyond Olejnik. Again the home players celebrated en masse. "To lose two goals from crosses is poor," said Ferguson. "But Sunderland deserved to win."

Tougher, much tougher, tests like ahead starting with the visit of Liverpool, Luis Suárez et al, on Sunday – when Ball is still expected to be in temporary charge – but at least Sunderland have finally remembered how to win. "Anybody's who's anybody wants to be considered for this job," said Ball when asked if he wants to succeed Di Canio. "But that's up to the board. Whatever they decide, I will go with it."