The European champions, who are trying to rebuild after failing to qualify for a second straight World Cup, now travel to Wolverhampton to face bottom side England on Saturday before taking on Germany in Moenchengladbach three days later.
"It was a good performance, especially in the first half. Their goal made us a little bit fearful but it was a match which we should have won by a bigger margin," Mancini told RAI.
"This is a team full of young players who need to work a lot, we've got a long way to go."
Mancini selected a host of players who played little or no role in Italy's triumph at Euro 2020, the glow of which has dimmed after they were dumped out of this year's World Cup in Qatar by North Macedonia in March. Before Tuesday's match Italy had won just three of the 11 fixtures played since being crowned kings of Europe at Wembley in July, one of those a meaningless win in Turkey days after their play-off defeat to North Macedonia. The hammering Italy took at the hands of Argentina in last week's 'Finalissima' in London seemed to confirm a deep crisis.
But a youthful XI showed signs of life in Saturday's 1-1 draw with Germany and they took that momentum to Cesena, frequently playing with the style and verve which characterised Mancini's team before the Euro.
Barella opened the scoring in the 30th minute, firing home his eighth Italy goal via two slight deflections after great work down the left from Leonardo Spinazzola, who seems near to top form after nearly a year out injured.
Pellegrini almost laid on teen talent Wilfried Gnonto moments before the break when he scampered in behind and rolled an inviting pass across goal which was well cut out by Loic Nego.
However the Roma captain then doubled the hosts' lead seconds later, making no mistake when meeting Matteo Politano's pull-back to net his second Italy goal in four days.
Politano crashed a shot off the bar nine minutes after the break and six minutes later Hungary were back in the game when Roma's Mancini, who up to that point had been one of Italy's best performers, turned in substitute Attila Fiola's low cross under no pressure.
From there the away side pushed for a leveller as the home side visibly tired, but beyond a couple of speculative efforts couldn't build on their opening win over England.