I have tried many free backup solutions, and recommended several here, but after another recent round of comparison, I have happily spent US $19.99 to purchase bvckup2.
I bought it primarily for its speed, since I have large amounts of data. I particularly like how it only copies changes.
But what interests you is
Backup of locked files
Use Windows shadow copying to backup files that
are locked for exclusive use by running programs such as web browsers
or email clients.
There is a 2 week fully functional free trial, and you can also download the last beta (forever), but it is well worth the $19.99. Visit the website and read all of the features.
[Update] I recently had cause to purchase tow more licenses, wrote to Alex, the author & he gave me a 20% discount code. Yesterday evening, one license stopped working as I had swapped motherboards and added a few new drives to my server, thus giving a different "hardware fingerprint" and invalidating the license. I emailed him and got an answer within 2 or 43 minutes, which is typical of his response time on support (if his time zone is awake when mine is).
A great product, with great support, which does exactly what the question asks for - and oh so very much much more..