Two goals and a Bruno Martins Indi red card in the first 10 minutes set up the hosts for a comfortable victory against a side undergoing changes with new boss Guus Hiddink
Conte – tasked with leading Italy into a new generation after Cesare Prandelli's post-World Cup resignation – could not have hoped for a better start in Bari.
Immobile fired them ahead after just three minutes when a long Leonardo Bonucci ball caught the Netherlands defence napping.
And worse was to come for the visitors - beginning their own new era under Guus Hiddink, who also coached them between 1994 and 1998.
Bruno Martins Indi was sent off in the ninth minute for bringing down Italy debutant Simone Zaza in the penalty area and De Rossi coolly converted the resultant spot-kick.
Italy had numerous chances to extend their lead, particularly in the first half, but Netherlands goalkeeper Jasper Cillessen did well to keep them at bay before the contest fizzled out in the latter stages.
Conte - widely regarded as one of the best coaches in Europe after leading Juventus to three successive Serie A titles - will be encouraged not only by the result, but also by the impressive performance from his side.
Granted, Martins Indi's early dismissal ensured the match was played on Italy's terms, but their bright interplay and attacking verve is sure to have buoyed Conte ahead of the nation's first Euro 2016 qualifier in Norway on Tuesday.
Immobile quickly gave Italy the lead when Bonucci's hopeful punt split Martins Indi and Stefan de Vrij.
Cillessen came charging out of his goal, but Immobile had enough time to round the Netherlands goalkeeper before tapping in with his left foot to open his account for Italy.
Wesley Sneijder curled an effort over at the other end but it was a rare sight of goal for the Netherlands, who quickly found themselves down to 10 men when Martins Indi - the last man - brought down Zaza.
De Rossi rolled his effort into the bottom-left corner as Hiddink swapped Jeremain Lens for defender Joel Veltman in a bid to stem the flow of goals, and while that was successful, the change did little to halt Italy's momentum.
Zaza should have marked his debut with a goal in the 20th minute, but was denied by a great Cillessen save after Immobile unselfishly centred, while the goalscorer shot just over after combining well with his strike partner.
Immobile missed the target again when played in by Zaza on the stroke of half-time and the latter's quest for a goal continued after the break when he shot into the side netting.
Robin van Persie fired just wide for Netherlands after Andrea Ranocchia's poor headed clearance, but like Sneijder's first-half effort, it was a rare foray forward for the visitors.
Italy still controlled the match but were restricted to long-range efforts - Bonucci and Zaza both tested Cillessen from distance, while Mattia Destro headed over - in the second half, as a flurry of substitutions saw the match peter out.
Two goals were more than enough for Italy, though, as the Netherlands received a wake-up call ahead of their opening Euro 2016 qualifier against Czech Republic on Tuesday.