Fayyad is former World Bank employee who has good relations with the West, and Marwan Bishara, Al Jazeera's senior political analyst, said that he is thought of as a safe pair of hands.
Bishara said: "He is not a strong man in Palestinian society and is not known for his charisma or vision. He is known as a correct accountant, not a reckless investment banker."
The appeal for dialogue from Meshaal, the chief of Hamas's political bureau, came at a news conference in the Syrian capital Damascus, where he lives in exile, within hours of an Arab foreign ministers' meeting in Cairo to discuss the crisis.
"What is needed now is to deal with the Palestinian schism. Hamas is for Arab sponsorship of a dialogue in the Palestinian national interest.
"If you ask me Fatah and Hamas are the same: both seek power and both want to obtain it through violence"
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He said Hamas had no option but to use force to wrest control of Gaza
in fighting that killed dozens of Palestinians.
"The lack of security drove the crisis toward explosion. What happened in Gaza was a necessary step. The people were suffering from chaos and lack of security and this treatment was needed."
Meshaal called for the Palestinian security forces to be reorganised and said elements in Fatah that he did not want to identify were to blame for the lack of security in Gaza.
"We need to restructure the Palestinian security apparatus to be a national force chosen according to merit and not on a factional basis," he said.
Before the announcement of Fayyad as his replacement on Friday, Haniya said he remained open to dialogue with Abbas.
He said: "I still stress that the door is open to restructure Palestinian relations on the basis of national values."
Earlier Haniya said that the "unity government" would continue to function despite being sacked by Abbas.
He called his dismissal and the declaration of a state of emergency on Thursday "hasty".
Nour Odeh, Al Jazeera's correspondent in Gaza, said: "We know that he can carry on leading the government in the Gaza Strip with the Hamas ministers who are in the Gaza Strip.
"We do not know how they will be able to fund the government, how they will bring in food and supplies or how they will be able to have any influence in the West Bank."