Heard, 29, and 52-year-old Depp appeared in a court in Australia on Monday, where the actress avoided conviction for bringing their Yorkshire terriers into Australia after pleading guilty to making a false statement on her immigration card about the dogs.
The actress received a one-month good behavior bond. If she breaks the bond she will be ordered to pay Australian dollars 1,000.
In the apology video posted to the Australian Government Department of Agriculture and Water Resources' Youtube channel, the couple expressed remorse and instructed other visitors to respect Australian laws.
Depp added: "It has to be protected."
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Later in the video, Depp appeared to make an indirect reference to the couple's "firm" treatment from Barnaby Joyce, the Deputy Prime Minister of Australia, and his threats to put the dogs down if they were not returned to the US.
"And Australians are just as unique, both warm and direct. When you disrespect Australian law they will tell you firmly," Depp said.
Australia is known for its stringent quarantine laws designed to keep disease out of the country. Dogs arriving from the US must spend 10 days in quarantine.
"The Danish Girl" actress brought the dogs into Australia on Depp's private jet on a visit while he was filming "Pirates of the Caribbean: Dead Men Tell No Tales," without permits or a period in quarantine, flouting Australia's strict animal import laws.
Heard's lawyer Jeremy Kirk has submitted a video to the court of her expressing remorse and respect for Australia's bio-security laws.
"Ms Heard has made a video, the significance of which will become apparent... It relates to an expression of remorse and recognition of the importance of compliance with Australia's border protection laws," said Kirk.