Costa, a Brazilian-born forward who was recently naturalized by Spain, played the full 90 minutes against Italy on Wednesday night and helped his team team win 1-0.
"Costa made a great effort," Del Bosque said to Spanish TV station Telecinco. "Little by little, he will adapt to this team's dynamic."
Costa added: "I'm very happy to have made my debut with Spain with a victory. I know the first games are not easy and I believe I can give a lot more.
"I will continue to work hard in order to give all that I can to the national team.
"The (Spain) coach has a lot of confidence in me and I'm going to try and play the best that I can for Atletico (Madrid) in order to arrive well to the national team."
Del Bosque felt his side were worthy winners.
"We have played a good game," he said. "Considering it was a friendly, we've had very good spells.
"Italy had their chances but we had long stretches of great football.
"We played with vitality and with speed."
Barcelona winger Pedro, who scored the only goal of the game, believes the victory will give the world champions a boost in confidence ahead of this summer's World Cup.
"We have beaten Italy in recent times but it's always a hard rival to beat," he said. "This victory allows us to gain confidence."
Italy coach Cesare Prandelli was disappointed with his team's overall display.
"At this moment of the season the fitness levels compared to Spain are almost embarrassing," he said. "We need to work hard, as to play with a high tempo we need intensity in our running.
"We physically weren't sharp enough.
"At this time, with these fitness levels it's impossible to judge the difference in quality between us and Spain.
"Quite simply, they always got to the ball first."