I would agree with your solution. Reading the file one line at a time should avoid the overhead of reading the whole file into memory at once, which should make the application run quickly and efficiently, primarily taking time to read from the file (which is relatively quick) and parse the lines. The one note of caution I have for you is to watch out if you have embedded newlines in your CSV. I don't know if the specific CSV format you're using might actually output newlines between quotes in the data, but that could confuse this algorithm, of course.
Also, I would suggest batching the insert statements (include many insert statements in one string) before sending them to the database if this doesn't present problems in retrieving generated key values that you need to use for subsequent foreign keys (hopefully you don't need to retrieve any generated key values). Keep in mind that SQL Server (if that's what you're using) can only handle 2200 parameters per batch, so limit your batch size to account for that. And I would recommend using parameterized TSQL statements to perform the inserts. I suspect more time will be spent inserting records than reading them from the file.