The Briton lost his WBA and IBF belts in a hotly-disputed loss to Peterson in the Washington fighter's hometown last December.
Khan raised a number of grievances surrounding the result and was further vindicated when Peterson tested positive for synthetic testosterone before their scheduled May rematch.
Golden Boy Promotions chief executive Richard Schaefer told Press Association Sport the WBA informed him that Khan has been given his title back and will put it on the line in a unification bout with WBC champion Danny Garcia on Saturday.
"Justice has been done," Khan told Press Association Sport.
"I'm glad the WBA are reinstating me as champion, it means I walk into this fight as world champion.
"There is a chance to win the WBC belt as well but not only that, the Ring magazine title will be on the line as well.
"It means this fight will really show who is the best fighter in the 140lb division."
Both the World Boxing Association and International Boxing Federation have taken their time considering what action to take.
Schaefer claimed recently the WBA had decided to strip Peterson of the belt and put it on the line, but the WBA, true to form, made no public statement.
However, Schaefer revealed they have gone a step further by not simply declaring the title vacant but, crucially, awarding it to Khan. Khan will receive the hollow title of 'super champion'.
Peterson admitted to using the testosterone - apparently for medical reasons - before their controversial encounter, calling into question the validity of his win.
While the result has not been overturned and seems unlikely to be in the future, the WBA's decision adds weight to Khan's belief that the loss should be regarded as null and void.
A statement from the WBA said: "The World Championships Committee of the World Boxing Association officially announced today its decision to reinstate the British boxer Amir Khan as super champion who may be able to unify title with Danny Garcia on Saturday July 14.
"This decision was taken after the positive doping result of the American Lamont Peterson, confirmed by the medical experts.
"The WBA rules that states that no boxer has tested positively for prohibited substances can be rated, retain a title, or be permitted to fight in a sanctioned bout for a period of no less than six (6) months from the date of the positive test has been enforced.
"Given the circumstances, Peterson's status is champion in recess and it is subject to revision."
Schaefer was also overjoyed at the WBA's decision and hopes Khan will prove himself as the world's best light-welterweight on Saturday.
"I got confirmation today that the WBA have reinstated Amir Khan as their super champion and so he will go into the fight defending his title," Schaefer told Press Association Sport.
"Danny Garcia will be defending his WBA belt and we've just learnt that the Ring magazine are putting their belt on the line for the winner of this fight - all three belts will be on the line.
"It's great news. And it's great news for both fighters as the winner of this fight will be recognised as the best 140-pounder in the world."