The Palestinian, Abdel Fattah al-Sharif, 21, was one of two men accused of stabbing another Israeli soldier in the West Bank city of Hebron in March.
The second Palestinian suspect had already been shot dead.
The case against Azaria has divided opinion among the Israeli public, politicians and members of the military, and focused attention on the military's code of ethics.
The shooting came at the height of a wave of Palestinian stabbing and ramming attacks, amid criticism that Israeli soldiers were too quick to pull the trigger in response, often killing the assailants. Human rights organizations say Israeli soldiers rarely face discipline for violence against Palestinians.
Azaria became the lightning rod for that criticism.
Hundreds of supporters protested on Azaria's behalf outside the Tel Aviv courtroom where the trial was held, and senior politicians have called for him to be pardoned.
Israeli Prime Minister Benjamin Netanyahu on Wednesday added his voice to those in favor of a pardon.
On his Facebook page, Netanyahu wrote, "(T)his is a difficult and painful day -- first and foremost for Elor, his family, many citizens and parents of soldiers, among them me."
But the Israeli minister of defense said that the military would respect the verdict even if there were those who didn't like it.
Azaria was serving in Hebron, a majority Palestinian area in the West Bank, when he shot Sharif. The Israeli soldier who had been attacked suffered a minor injury.
According to the Israel Defense Forces, Azaria arrived at the scene about 11 minutes after the attack, cocked his rifle and shot Sharif in the head, killing him.
Azaria denied the charges against him.
But a three-member panel of military judges was unanimous in finding him guilty of manslaughter and improper conduct.
At the heart of the case was the motivation for the soldier's actions. Azaria's defense argued he felt his life was in danger, while the prosecution contended he was motivated by malice and vengeance.
In a lengthy explanation before the verdict was delivered, the judges rejected all the defense arguments.
The head judge, Col. Maya Heller, said that Azaria was an "unreliable" witness and that his defense witnesses were also problematic. She called the shooting "needless."
Azaria's attorneys have promised to appeal the conviction.
About 50 people packed Courtroom A inside the heavily guarded Ministry of Defense as they waited for the verdict to be read.
On one side of the aisle sat Azaria's family. On the other sat the media. Four members of the prosecution, dressed in uniform, waited across from Azaria's defense team, dressed in suits.
Family members gave Azaria reassuring hugs and pats on the back. His mother held him tightly.
Emotional scenes followed when the verdict was finally announced. After the judges walked out, Azaria's mother screamed, "You should be ashamed of yourselves."
Protesters chant, 'Our soldier, the hero'
Across the street from the Ministry of Defense compound, several hundred supporters gathered behind a police security barrier.
They danced and cheered, shouting, "Our soldier, the hero."
Some waved Israeli flags and homemade signs that read, "Today: Azaria Tomorrow: Your son." Other signs said, "The nation is with you!"
The IDF said in a statement that it respected the court verdict and would study its implications, while giving Azaria and his family all the support they need.
"The IDF judicial system will continue to act towards revealing the truth, with independence and professionalism, in any case of suspected wrongdoing by the IDF and its soldiers," the statement said.
But others, mostly politicians on the right, have argued that Azaria's actions were justified and that he should never have faced prosecution for killing a terror suspect.
Naftali Bennett, the Israeli education minister and Jewish Home party leader, called for an "immediate" pardon in a post on Facebook, saying the process had been "contaminated from the start."
Miri Regev, Israel's culture and sports minister, said she would "work to pardon" the convicted soldier. "This was a trial that should never have begun," Regev, a former brigadier general and spokeswoman for the IDF, wrote in a Facebook post.
However, human rights groups hailed the verdict.
In a statement, Amnesty International said the conviction was "a rare occurrence in a country with a long record of using excessive and unwarranted force," and that it "offers a small glimmer of hope amid the rampant impunity for unlawful killings" in the Palestinian territories.
Sari Bashi, Israel advocacy director at Human Rights Watch, said, "Today's conviction is a positive step toward reining in excessive use of force by Israeli soldiers against Palestinians."
Rights organization Adalah -- the Legal Center for Arab Minority Rights in Israel said the conviction was "the exception to the rule and stands unique in the long history of impunity enjoyed by Israeli armed forces."