Yes. The due date is for that statement. The additional charges are now part of the next statement and will have their own due date. As long as you pay the statement balance by the due date for that statement you'll avoid finance charges.
Just be aware that some cards have a limit to how many payments can be made in a certain time frame so don't leave both payments until the last minute. One of my cards would only allow a payment every 6 days (or something weird like that) but others will let you set up a payment every day so I'd just double check your timing works or you may need alternate arrangements. (push a payment or maybe even mail a check
)