Bypass escaped double quote
Based on your description this should work:
The " will be escaped as \", thus resulting in \", which escapes the \, but not the ".
To prevent this, you would at the very least have to also escape \ as \.
Bypass escaped double quote and double backslash to single backslash
Based on your comment, I'm assuming the actual relevant parts of the filtering work like this:
" -> \"
\\ -> \
This isn't secure either. An injection might look like this:
" will be escaped as \" in step 1 which leads us to \\", which is then transformed to \" in step 2. You can't use double quotes in the injected string, but that's not a problem as XSS with single quotes or completely without quotes is perfectly possibly.
To secure this the double backslash would need to be escaped, not transformed to a single one, which would give us \\\\\" with the above described injection, which is safe.