The missiles were fired 30 seconds apart and help explain a mystery as to why the plane's transponder was not working -- it was disabled by the first strike, before being hit by a second, the Times said.
The Ukraine International Airlines plane was brought down shortly after takeoff on Wednesday, killing all 176 passengers and crew on board.
Tehran had for days denied Western claims based on US intelligence that the Boeing 737 had been downed by a missile.
It came clean on Saturday when Revolutionary Guards aerospace commander Brigadier General Amirali Hajizadeh acknowledged a missile operator had mistaken the plane for a cruise missile and opened fire independently.
The blurry footage shows the plane on fire and circling back to Tehran's airport, the Times said. Minutes later, it exploded and crashed.
The footage was shot from a rooftop in Bidkaneh, a village four miles from an Iranian military site, the Times said.
Hundreds of angry protesters, most of them student, have taken to the streets in the wake of the tragedy, chanting slogans against the Islamic republic.
Iran announced its first arrests over the accidental shooting on Tuesday, without naming who or how many people had been detained.