He read the verdict on Thursday to cheers from the courtroom audience in the country's northern Tucuman province, where the victim served as senator.
Government estimates say anywhere from 13,000 to 30,000 people were killed in a crackdown on leftist activists during Argentina's seven-year dictatorship.
In a statement, Argentine Human Rights Secretary Eduardo Luis Duhalde praised the verdict.
"We are witnessing the end of impunity and the restoration of the truth, memory and justice," he said.
Bussi, 82, will be allowed to remain under house arrest because he is over 75.
That decision enraged human rights activists outside the courtroom, who threw rocks and swatted at police with sticks.
Police in riot gear used tear gas to restrain the angry crowd.
Menendez, who is 81, will remain in prison in another province, where he was convicted in July of human rights violations.
In his final words to the court, Menendez defended the actions of the armed forces during the dictatorship.
"Argentina flaunts the dubious merit of being the first country in history to judge its victorious soldiers," he said.
"They call the operations of the armed forces illegal repression."
Bussi cried as he said: "I've never seen or done anything to Mr. Vargas Aignasse in my entire life."
After democracy was restored in 1983, Bussi was accused of kidnappings and assassinations, but an amnesty law prevented him from being prosecuted.
He was elected mayor of the capital of Tucuman in 2003, then arrested three months later for the disappearance of Vargas Aignasse.
In 2005, Argentina's Supreme Court struck down sweeping amnesties that had shielded hundreds of former officers from prosecution.