Netanyahu told Bolsonaro he was certain his election "will lead to a great friendship between our peoples and the tightening of links between Brazil and Israel".
Mr Moro was floated as a possible presidential candidate in the run-up to this year's election.
Sergio Moro is wildly popular among conservatives and loathed by many on the left, so his decision to join the incoming administration will feed the suspicion of many Brazilians that the judge was politically biased in jailing ex-President Luiz Inacio da Silva, a conviction that forced the poll-leading leftist out of the presidential race.
"These are provocative and illegal steps that will only destabilise security and stability in the region", said Hanan Ashrawi, a member of the Palestine Liberation Organisation's executive committee.
Moro said he hoped to "consolidate progress against crime and corruption".
Liberal economist Paulo Guedes - Mr Bolsonaro's pick to head an economy "super ministry" combining finance, planning, industry and trade - vowed immediately after the election to accelerate the pension reform, cut spending and privatize state enterprises.
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But on Wednesday night, a Bolsonaro ally said the president-elect had not yet made up his mind regarding the merger. That figure dropped sharply after he sentenced Lula, who is still remembered fondly for reducing inequality during his 2003-2011 presidency.
Bolsonaro will be sworn in on January 1, 2019. At his first formal news conference on Thursday, he gave fresh details of his agenda.
He did, however, back down from one controversy on Thursday: a plan to merge the agriculture and environment ministries.
Bolsonaro also said that a proposed deal between plane makers Boeing Co and Brazil's Embraer that requires government approval is "going through", without elaborating.
Asked in an interview with Israel Hayom published yesterday if he would move Brazil's embassy, as he had indicated during his campaign, Bolsonaro said Israel should decide where its capital is located.
Bolsonaro -dubbed "the Brazilian Donald Trump" -has expressed agreement with the USA on several foreign policy issues, criticizing the leftist regime of Venezuela and China's economic power.