"We are delighted," Klinsmann said. "It is looking positive after this.
"We absorbed a lot of pressure and I was happy with the performance."
The 1990 World Cup winner was confident his side was ready for such a performance.
"You could see the energy in the changing room," he said.
The hosts are also set to receive a further boost for their next group match against Poland with the return of captain Michael Ballack.
"We hope Michael will be back for the Poland game," he added.
Meanwhile, Germany and Costa Rica also set a new record for the amount of goals scored in the opening game of the tournament.
The hosts' tally was the first time a side has scored more than 3 goals in the opening match of the tournament since 1962.
Between 1966 and 1978 not a single goal was scored in the opening match, while between 1982 and 1994 no side scored more than one goal in the curtain raiser.
Germany’s efforts were also the first time since 1962 that a side has won the first match of the World Cup by two goals.