Barca's hopes of leaving Andalusia with a much-needed win were dealt a blow when De Jong received a second yellow card in the 65th minute for following through on Alfonso Espino after initially winning the ball.
The Catalan giants' woes were compounded deep into stoppage time when Koeman was also ejected for his protestations towards referee Carlos del Cerro Grande.
The stalemate means Barca hasdrawn three of its opening five LaLiga games this season to sit seven points adrift of leader Real Madrid, although Carlo Ancelotti's side has played a game more.
Koeman's dismissal came after a frantic period that saw two balls in play at the same time, with the Dutchman claiming he was given his marching orders by Cerro Grande for alerting him to the situation. Sergio Busquets was booked for kicking the rogue ball towards the Cadiz player in possession.
"It was not because of nervousness, it was because I told the fourth official that there was a second ball on the field and the referee had to stop the game. Everyone except the referee saw it," Koeman said. "They expel you for nothing, in this country they expel you for nothing. I have asked the referee as normal. Attitude? We leave it because it is not my problem."
The goalless draw means Barca has failed to score in two of its past three games across all competitions – as many as in its previous 41 matches.
Meanwhile, its tally of eight goals scored at this stage of the season is its worst since the 2003-2004 season when it scored just five in its opening five matches.
Koeman's future has been the subject of intense speculation in recent weeks and the Dutchman is aware that is unlikely to subside in the coming days.
"I do not talk about my personal situation," he said. "It seems that if I win I continue and if I lose the club has to find another coach. It is for you [to talk about].
"The president [Joan Laporta] was at the hotel, and we greeted each other shortly before leaving. Surely there will be more days to talk about this issue if the club wants to talk."
Barca is next in action against Levante at Camp Nou on Monday (AEST).