The death of Savimbi has fueled hopes for an end to a civil war that has dragged on since Angola's independence from Portugal in 1975, and Bush said he urged dos Santos at a White House meeting "to move quickly toward achieving a cease-fire."
Dos Santos said he wanted a cease-fire in place "as soon as possible" and was committed to holding democratic elections "as soon as we have security."
But he put the onus on UNITA forces to offer a "sign of goodwill" to the government by declaring a "cessation of hostility."
UNITA rebels killed nine people and wounded 15 in an attack in central Angola on Monday.
"It will depend on the political will of those who are fighting," dos Santos said.
The 67-year-old Savimbi was confirmed dead by his National Union for the Total Independence of Angola (UNITA) movement on Saturday after state television showed his bullet-riddled body. He had been shot dead by government troops.
The United States, along with apartheid South Africa, gave support to Savimbi in the 1970s and 1980s.
"President dos Santos has it within his power to end 26 years of fighting by reaching out to all Angolans willing to lay down their arms. Angolans deserve no less," Bush said in a statement released by the White House.
"And we agreed that all parties have an obligation to seize this moment to end the war, and develop Angola's vast wealth to the benefit of the Angolan people," he added.
UNITA vice-president Antonio Dembo has taken over the interim leadership, a movement spokesman said in Lisbon.
Diplomats described Dembo, a military commander who fought alongside Savimbi since 1975, as a moderate who could steer UNITA toward a cease-fire and transform it into a formidable political party.
After his meeting with Bush, dos Santos said the UNITA movement, absent Savimbi, was "very weak" and brushed aside questions about his commitment to democratic reforms.
"We're going to proceed with elections as soon as we have security," he told reporters. "We are in the process of democratization."
After meeting earlier this week with Portuguese President Jorge Sampaio, dos Santos said elections could be held within 18 months to two years, but it depended on a cease-fire and demilitarization of UNITA this year.