"The death sentence will be carried out on a group of jihadists, starting with Rishawi, as well as Iraqi Al-Qaeda operative Ziad Karbuli and others who attacked Jordan's interests," said the official, speaking on condition of anonymity.
King Abdullah II, who was visiting Washington as the video came to light, recorded a televised address to his shocked and outraged nation.
The king, who was once in the military himself, described First Lieutenant Maaz al-Kassasbeh as a hero and vowed to take the battle to Islamic State extremists, who have executed several captives on camera in recent months, provoking worldwide revulsion.
"Jordan's response will be earth-shattering," Information Minister Mohammed Momani said on television, while the army and government vowed to avenge the pilot's murder.
"Whoever doubted the unity of the Jordanian people, we will prove them wrong."
US President Barack Obama, who hosted Abdullah in a hastily organised and brief Oval Office meeting, led widespread international condemnation of the latest graphic murder, decrying the "cowardice and depravity" of the Islamic State group.
"The president and King Abdullah reaffirmed that the vile murder of this brave Jordanian will only serve to steel the international community’s resolve to destroy ISIL," a National Security Council spokesman said after the pair met.
The Obama administration had earlier reaffirmed its intention to give Jordan $3 billion in security aid over the next three years.
- 'Unforgivable' -
Kassasbeh was captured in December when his jet crashed over northern Syria on a mission that was part of the US-led coalition air campaign against the jihadists.
Jordanian state television suggested he was killed on January 3, before IS offered to spare his life and free a Japanese journalist in return for Rishawi's release.
The White House would not speculate on whether the video was released to coincide with Abdullah's visit to Washington.
British Prime Minister David Cameron called the murder "sickening," while UN chief Ban Ki-moon labelled it an "appalling act".
Japan's Prime Minister Shinzo Abe condemned it as "unforgivable".
The death marks a further escalation of Islamic State's execution shock tactics after a series of brutal hostage beheadings.
The highly choreographed 22-minute footage shows Kassasbeh at a table recounting coalition operations against IS, with flags from the various Western and Arab countries in the alliance projected in the background.
It then shows Kassasbeh dressed in an orange jumpsuit and surrounded by armed and masked IS fighters in camouflage.
It cuts to him standing inside a cage and apparently soaked in petrol before a masked jihadist uses a torch to light a trail of flame that runs to the cage and burns him alive.
The video also offered rewards for the killing of other "crusader" pilots.
- 'Act of belligerence' -
Shiraz Maher, from the International Centre for the Study of Radicalisation at King's College London, described the footage as "simply the most horrific, disgusting thing I have seen from Islamic State in the last two years".
"They clearly want to make a real point. This is the first individual whom they have captured who has been directly involved with the Western coalition in fighting IS. It is different from the aid workers... This is an act of belligerence.
"Every time you think they cannot commit anything worse -- they open up another trapdoor."
The release of the video came after IS beheaded two Japanese hostages within a week.
The Islamic State group last year declared an Islamic "caliphate" as it rampaged across Iraq and Syria in a brutal offensive of executions and forced religious conversions.
The United States, Jordan and other nations responded with "Operation Inherent Resolve," an air-led campaign to pummel the jihadist group.
Kassasbeh's plane was the first loss of an aircraft since the US-led coalition launched strikes against IS last year.
US Central Command said that the group still had the ability "to conduct small-scale operations," despite months of air strikes.
But, it said, "their capacity to do so is degraded and their momentum is stalling."
Attacks have hit the IS group's "ability to command and control forces; recruit, train and retain fighters, produce revenue from oil sales, and maintain morale."