Hi Mate,
You can include Azure Automation Runbooks in your fail-over plan which is written in powershell. It will help you to update the DNS.
Please follow below urls.
Add Azure Automation runbooks to recovery plans
https://docs.microsoft.com/en-us/azure/site-recovery/site-recovery-runbook-automation
Azure Site Recovery Automation Runbooks
https://github.com/Azure/azure-quickstart-templates/tree/master/asr-automation-recovery
Regards
Chinthaka