Arnold and his players are under fire after kicking off the third phase of Asian qualifying with a 1-0 home loss to Bahrain and a scoreless draw away to Indonesia.
Johnson, however, said Arnold had the FA's full backing.
"Graham, he's a great coach, and we're right behind him," Johnson told reporters on Friday.
"He's got a bit of work to do between now and when the team comes back to Australia, but we're not panicking in any way."
Australia lie fifth in Group C, five points adrift of leading Japan, with eight matches left in the phase.
The top two teams qualify automatically for the World Cup, though the third and fourth-placed teams have chances to qualify in subsequent phases.
Australia next host bottom-placed China in Adelaide on October 10.
"Of course we want to see improvements based on this window because we're disappointed with the results," said Johnson.
"We're not where we want to be.
"What is required at the moment is cool and calm leadership. And we put our faith in Graham to get the team back on path when we play against China."
The Matildas
The Australian women's team has also struggled of late, suffering an early exit from the Paris Olympics after bringing ambitions of a first soccer medal for the country.
With coach Tony Gustavsson departing after Paris, Johnson said he expected an interim coach to be named within 10 days while the search for a permanent replacement goes on.
The Matildas are slated to play a friendly away to Germany on October 28.
Johnson said there was no timeline to land a permanent coach for the World Cup semi-finalists but "the sooner the better".
"At the same time we want to make sure we get the appointment right because we're setting up - as we did four years ago - for a whole new cycle and we want to ensure that this is a long-term appointment," he said.
"And it's an important appointment, the Matildas are a great football team and they're also a big brand, they're a big part of the business, so we've got to get this decision right."
