FBF president Rolando Lopez told reporters that FIFA had revoked its decision to award 3-0 walkover wins to Chile and Peru after Bolivia fielded Paraguay-born Nelson Cabrera, who was not eligible to play.
Bolivia had beaten Peru 2-0 and drawn 0-0 with Chile.
"The country can remain absolutely at ease," said Lopez. "The points have been returned to us, everything is as it should be."
He quoted a fax which he said had been sent by FIFA allegedly telling the FBF to "dismiss the content of the decisions which were notified on November 1".
Shortly afterwards, FIFA denied that version of events.
"The sanction on Bolivia is maintained. The version circulating about the return of the points to Bolivia is incorrect," world soccer's governing body said on Twitter.
Under FIFA rules, players can only play for an adopted country if they have lived and played there for five years, and provided they have not featured in a competitive game for another national side.
Cabrera played for Paraguay in a friendly, which would not have been a problem, but only moved to Bolivia four years ago.
The decision was a boost for Chile as the two extra points took them up from seventh to fifth in the South American group and ahead of Argentina on goal difference. Both teams have 16 points from 10 games.
Peru's outside chances were also helped as they moved onto 11 points, in eighth place.
The top four teams qualify directly for the 2018 World Cup in Russia and the fifth enters a playoff against a side from another continent.
(Writing by Brian Homewood in Zurich; Editing by Toby Davis)
