Lionel Messi could not be signed to a new deal because of spiraling costs at the Catalan club and, with no signs of improvement, Barca risked not being able to register new signings Memphis Depay, Eric Garcia and Aguero.
However, after Pique agreed to a salary reduction and deferral of payments, Ronald Koeman's side confirmed they were able to register Depay, Garcia and Rey Manaj.
Alba and Busquets have now, too, agreed to alterations to the terms of their contracts.
Both players will collect reduced wages for this season and deferred payments for the rest of their deals, with also Sergi Roberto reported to be negotiating changes to his terms.
The agreements with Busquets and Alba mean cash-strapped Barca can now register Aguero, who is expected to be out injured until the middle of October, while avoiding complications with financial fair play regulations.
Left-back Alba created the second most chances at the club last term in LaLiga (53), while Busquets led the way for tackles and interceptions while also completing the second most passes.