The dog was held in the council's pound following the attack.
Communications and marketing manager Graham McKerracher said the decision to euthanise the dog was made following a meeting with its owner today.
Mr McKerracher said while the situation was distressing for all involved, the DCC was responsible for the safety of its residents and enforcing dog control laws. It was taking action appropriate to the seriousness of this issue.
People could report aggressive dog behaviour, or attacks, to the council at any time, he said. That allowed officers to respond to any community concerns.
The Dog Control Act 1996 prescribed two offences. The council would evaluate the case to determine whether the charges are laid under one or both.
As council was now formally seeking a prosecution, it would be inappropriate to make any further comment at this time, he said.