Puerta, 22, collapsed in the first-half and medics prevented him from swallowing his tongue.
But he collapsed again after going off and was given cardiac resuscitation before being taken to hospital.
He was placed in intensive care and doctors said on Tuesday his condition had deteriorated before his death.
Puerta collapsed after jogging back towards his own goal 35 minutes into the opening league game of the season at the Sanchez Pizjuan on Saturday.
Team-mates and medical staff rushed to his side and were seen preventing him from swallowing his tongue as he lost consciousness.
He was able to walk from the pitch but doctors said that he collapsed again in the changing rooms and had to be given cardiac resuscitation before being taken to the Virgen del Rocio hospital.
Hospital director Gimeno Guerrero said Puerta died of "multiple organ failure, stemming from prolonged cardiac arrest."
As a consequence, the league match against Osasuna that was scheduled for next Monday has been postponed.
There will also be a minute's silence in tribute of Puerta before every La Liga match next weekend.
A league statement said: "The Professional Football League wishes to pass on its most sincere condolences to the family of Antonio Puerta.
"It has also declared the next day of league play to be one of official mourning and asked all its affiliated teams to observe a minute's silence before the games."
European football's governing body said: "UEFA and the entire football family extend their sympathy and sincere condolences to the family of Antonio Puerta."