Yes, but be prepared to be delayed at check-in.
Australia's visas are fully electronic and are linked to your passport number. You are required to submit your new passport details (via Form 929 or ImmiAccount) when you renew your passport so that your visa can be linked to your new passport number.
When you check in to your flight, the check-in agent will verify your visa details directly with the Australian Government using your new passport number. If your passport details have not yet been updated, the system will not be able to find your visa and you may be denied boarding.
Therefore, you should bring your old and new passports when you check in. The check-in agent (if they are aware of the proper procedures) can manually telephone the Border Operations Centre to update your passport details if your Form 929 is not processed in time. A FAQ on this process can be found here.
You can check whether your visa has been transferred across by using VEVO with your new passport details.