Simone Zaza and Leonardo Bonucci were on target to give Italy a 2-0 win in Norway and make it a dream start for Conte, who replaced Cesare Prandelli after he resigned following the Azzurri's first-round exit at the World Cup.
The result in Group H also saw the Italians extend their impressive unbeaten record in qualifying to 41 games, stretching back eight years.
Meanwhile, a late goal from Vaclav Pilar saw Hiddink's Netherlands slip to a 2-1 defeat to the Czech Republic in Prague.
However, with the top two in each group automatically qualifying for the finals in France and play-offs for the third-placed teams the Dutch should still have no problem in advancing from Group A.
Pressure, though, will be growing on Safet Susic, who was thought fortunate to have survived in the Bosnian job after their disappointing first-round exit at the World Cup finals and a 2-1 home defeat to Cyprus in Group B on Tuesday did him few favours.
Both Susic and Terim could have done with someone of the star quality of Gareth Bale, for the Real Madrid forward came to Wales boss Chris Coleman's rescue in scoring a double, the second a marvellous free-kick 10 minutes from time, as they came from behind to edge minnows Andorra 2-1.
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