"I am going to fight the enemies of the humanity and the enemies of our homeland who are trying hard to erase its identity and destroy its gains and humiliate the Yemenis", said Major General Ahmed Ali Abdullah Saleh, Yemen's ambassador to the UAE and the former commander of the Republican Guard.
Speaking before a gathering of supporters, al-Houthi says: "We knew that they were deceived. we hadn't hoped for what happened".
US-backed Saudi coalition forces renewed their attacks on Sanaa in order to back up Saleh's supporters.
It was not immediately clear if the rebels would allow Saleh's family to hold a funeral later in the day.
A video provided to AFP by the rebels showed what appeared to be a dead Saleh with a severe head injury, his body wrapped in a floral-print blanket.
Yemen's President Abed Rabbo Mansour Hadi has urged Yemenis to unite against the country's Shiite Houthi rebels, describing them as "Iranian militias" and a "nightmare".
Global anti-poverty organization Oxfam praised Trump's action but called on him to do more to stop the bloodshed, including pushing for a ceasefire and ending arms sales to the Saudi-led coalition.
Saleh ruled Yemen for more than three decades until he was forced to resign following an Arab Spring uprising in 2011.
The killing of Yemen's ex-President Ali Abdullah Saleh by the country's Shiite rebels as their alliance crumbled has thrown the almost three-year civil war into unpredictable new chaos.
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However, close sources have denied the reports that Saleh was killed.
Saleh's former Houthi Shiite allies claim he was killed for "treason" for reportedly seeking to switch alliance to the Saudi-led coalition that has been fighting the Houthis for almost three years.
Saleh's slaying likely gives the rebels the upper hand in the dayslong fighting for the country's capital, Sanaa.
The fresh violence has increased fears for civilian victims of Yemen's war, which has claimed more than 8,750 lives since the Saudi-led coalition intervened.
Saudi Arabia and its allies struck a day after Saleh's son vowed to lead a campaign against the Houthis.
From the Saudi capital, Riyadh, where he has been in self-imposed exile for most of the war, Hadi tried on Monday to rally Saleh's allies to keep up the fight against the Houthis.
Clashes in the Yemeni capital have killed at least 234 people and wounded 400 since December 1, the International Committee of the Red Cross said on Tuesday.
Jamie McGoldrick of United Nations aid agency OCHA says civilians in Sanaa are "emerging from their houses after five days being locked down, basically prisoners", to seek safety, medical care, fresh water and other survival needs.