-replace uses regex for the search pattern and therefore must have regex-special characters escaped if they are to be matched literally. The \ is used for escaping.

$file = Get-Content C:\uniquefile1.txt -Raw
$file -replace '>>>11\^A', '>>>0111^A' | Set-Content C:\uniquefile1.txt

Select-String without the -SimpleMatch switch uses regex as well. I don’t know how that ever matched for you if the -replace operation failed.