The Brazilian-born striker responded to the jeers and insults from local fans at the Arena Fonte Nova yesterday by setting up Spain's opening goal of the tournament.
It was all downhill from there as the Dutch blew Spain away on the counterattack, with the Brazilian-born, Spanish national looking up at the scoreboard in disbelief with every goal conceded.
"On one hand I'm very happy for my debut but then, on the other hand, there is disappointment," Costa said. "You have to keep your head up high and rebound from it."
Costa won a penalty in the 26th-minute that Xabi Alonso converted a minute later to give Spain its lone bright moment of the Group B opener, a rematch of the 2010 final won by Spain.
But Costa was largely ineffective beyond that, his physical and gritty play unable to lift Spain, which came unstuck after Robin van Persie equalized just before halftime.
After the restart, Arjen Robben scored two goals, Van Persie added a second and Stefan de Vrij another as the Dutch dealt Spain its greatest competitive defeat since 1950.
"A game like this can happen," said Costa, who was jeered with every touch. "People should keep their faith in us and know we'll look to rally back."
Spain will probably have to win its second Group B match against Chile on Wednesday to advance to the elimination stage.
Costa will also have to raise his game, as his only impact was an ugly head butt to Netherlands defender Bruno Martins Indi shortly before coming off for Fernando Torres in the 62nd minute that was not spotted by the referee.
While Costa grew up in Brazil, he honed his skills in Europe, mostly at Spanish clubs Rayo Vallecano and Real Valladolid before breaking through at Atletico following Radamel Falcao's exit last summer.
He enjoyed a career-best season at Spanish champion Atletico Madrid, scoring 36 goals in all competitions to help the club to its first domestic league title in 18 years.
"We have to keep our heads high, we know the importance of the next two games," Costa said. "People should not think we're down and out, we'll raise ourselves."